DailyDirt: Math & The Mind
from the urls-we-dig-up dept
Math is simultaneously illusory and real. On the one hand, it’s a set of arbitrary symbols and chosen rules that can be modified and manipulated as we see fit, since it’s just a human invention; on the other hand, the patterns it identifies and describes have intrinsic reality and significance. The same could almost be said about the concept of the human mind — so where the two intersect, there are fascinating things to be learned:
- Recent mathematical models are changing the nature of the scientific conversation about consciousness, with one controversial theory suggesting our minds are ultimately non-computable. Could cognition be something other than just extremely advanced computation? [url]
- Computers are now producing mathematical proofs that are essentially beyond human comprehension, or at least beyond our realistic ability to do any double-checking. One recently computed proof involves 13 gigabytes of recorded calculations, or more text than all of Wikipedia. [url]
- Researchers are beginning to understand what happens in the brain to cause Savant syndrome after an injury. Jason Padgett developed the ability to visualize complex mathematical and physical problems after suffering a severe beating, and it’s possible that this ability lies dormant in everyone. [url]
If you’d like to read more awesome and interesting stuff, check out this unrelated (but not entirely random!) Techdirt post via StumbleUpon.
Filed Under: computation, consciousness, jason padgett, math, savant
Comments on “DailyDirt: Math & The Mind”
Re: Mathematics/non-computability of consciousness:
“Memory functions must be vastly non-lossy, otherwise retrieving them repeatedly would cause them to gradually decay”
Uhh, retrieving memories repeatedly DOES cause them to gradually decay.
science as mental immune system
I’ve worked my way through only a few pages of the “consciousness is non-computable” essay, and of the (apparently unpublished and unreviewed) paper it cites…
…but it sure looks like baloney to me.
Read it as meth and the mind at first
Consciousness Is An Illusion
Trying to assign meaning to consciousness tends to lead to paradoxes, like in those experiments where the part of the brain where a decision is made to do something lights up after the part that sends the signals to the muscles to do it?either the conscious mind is a time-traveller, or it simply believes it is in control when it is not.
The Zen Buddhists knew centuries ago that there is no real self, no soul, just the illusion of one.
Re: Consciousness Is An Illusion
Consciousness is nothing more or less than a kind of self-awareness. There is no paradox in it.
I think that the illusion that you’re thinking of is that the “part” of us that is conscious is the part that is completely in control — but that’s only tangentially related to consciousness itself.
The proof that consciousness is not illusory is that we have enough self awareness to wonder if consciousness is illusory. “I think therefore I am” applies here.
Brains non-computable? Whatever in the world would have given anyone the idea it would be?
This is just me talking here, but I would never have assumed anything else; and find myself astonished anyone did think otherwise.
Whether we are pursuing the Higgs boson or just discovering that bromine is the 28th element essential to life, we eventually find that nature has done it first and done it better.
Computable processes are those that can be reproduced with a Turing machine. Turing machines are fine for computers–we use them all the time–but very pitiful for real life processes that involve such minor niceties as mathematical chaos. But not only do life processes live with chaos, they thrive on it and (in many cases) use it to their advantage.
It is pure conceit to think we could reproduce a brain with a Turing-complete machine of any kind. When to reproduce it, we would have to reproduce the non-deterministic mathematical chaos that is intrinsic in all life.
Non-deterministic on an intrinsically deterministic machine? Not likely.
Re: that involve such minor niceties as mathematical chaos
But this ?chaos? is still mathematical?you said it yourself.
Re: Re: that involve such minor niceties as mathematical chaos
But the special bit about chaos is that although it’s mathematical and is also a form of order (chaos is not randomness), it’s also not computable except at a very high level.
Personally, I suspect that consciousness can be elicited from sufficiently complex computation devices. However, there are two problems with this hypothesis: it may not be testable (how do we know if anything outside ourselves is conscious or not?) and the consciousness will certainly not be similar to human consciousness (which is what most people are really talking about in these sorts of discussions). In fact, to tie that bit to the testability problem — such a consciousness may be so alien to us that we’d never even notice it.
Re: Deterministic non-determinism?
Ummm, Turing machines aren’t intrinsically deterministic. Not that it matters, since nondeterministic TMs are equivalent to deterministic TMs. Also, chaos theory is applied to deterministic dynamical systems, not nondeterministic ones.
Re: Re: Deterministic non-determinism?
People have a hard time understanding this. Predicting the orbit of one planet around one star is relatively simple and stable, but predicting three orbiting objects is chaotic can be intractable.
There is more than enough unpredictability to make humans seem non-deterministic, when in fact we are totally deterministic.
Re: Re: Re: Deterministic non-determinism?
“in fact we are totally deterministic.”
This remains an untested hypothesis, so it’s going to far to state it as fact. We could be totally deterministic, but we don’t know.
Even today, metaphysical notions of this kind still crop up.
There is really now no mystery about consciousness whatsoever.
The mystical notions that have arisen in the past are purely illusory, an inevitable result of approaching the question by introspection. This, of course, was the only option available to earlier philosophers and many still have trouble escaping from that trap with its inevitable recursive loops.
Is is sad to see articles such as this resurrecting the old mystical notions within a scientific context.
Today, although the details of nervous system function of ourselves or other animals is very far from complete, we have sufficient information to have a rough idea of the gross workings of these systems.
From evolutionary considerations we can also now see how the essentially navigational function we like to call “consciousness”, “self-awareness” “sense of agency” and so forth is bound to arise.
Most, if not all organisms must, in principle, have some degree of consciousness (self-awareness). Even if only as the locus of its sensory and effector interactions with the external world.
This, of course, includes such creatures as bacteria and plants. Here’s why:
From our understanding of biological evolution by natural selection it becomes quite clear that provision of a navigational feature that involves some degree of self awareness is required for an organism to interact optimally with its environment.
It is a measure of its fitness for the prevailing environment and subject to selection pressure accordingly.
Furthermore, from a quite different discipline, we now have an excellent understanding of functionally analogous computational systems. And the composite that we call the Internet has, even now, comparable processing power to the human mins and is rapidly becoming endowed with the semantic linkages required of our particular kind of consciousness.
With these new tools at our disposal we can now view the phenomenon in a truly objective way. And then the hocus-pocus surrounding this issue vanishes!
This topic is part of the broad evolutionary model very informal outlined in “The Goldilocks Effect: What Has Serendipity Ever Done For Us?” (free download in e-book formats from the “Unusual Perspectives” website).
The far more formal treatment I am at present working on “The Intricacy Generator: Pushing Chemistry and Geometry Uphill” deals with this subject (and also the nature of mathematics in more detail.
Computability, deterministicity and computational tractability are three different things. (And for what it’s worth, I know of no persuasive argument associating chaotic systems with conscious systems in any way.) So please, anyone who wants to argue about these things, know what the terms mean and don’t just mix them together in a word salad.
Re: order, order
This this this. A handful of Google searches and a few minutes of reading is usually enough to get your bearings on subjects and terminology. Worst case is you find yourself woefully ignorant, which can be more useful than confirmation.
As for whether or not consciousness is computable, I am very leery of any argument that asserts (explicitly or implicitly) some fundamental difference between our brains and a general computing paradigm (ex. Turing machines, the Lambda calculus). I think that every single instruction or ‘step’ ever performed by any human-made computing device can be done by a human. That isn’t to say that computers can do everything we can, far from it! Rather, I think it shows that our minds or consciousnesses encompass every possible elementary operation for every computing machine conceived by humans, real or imagined.
TL;DR : Our minds are a set that includes the entirety of any model of ‘universal’ computation we’ve successfully described.
I’m going to go out on a limb and claim that no problem shown to be uncomputable for a universal model of computation is computable for our minds. I think if this were not the case, it would be obvious that the ‘universal model’ of computation was lacking, and could be extended in some fashion by the manner in which a human mind showed an uncomputable problem to be computable. I suspect that a closer reading of Godel’s incompleteness theorems could both support this and bite me in the ass.
Basically, the fact that we have always been able to do things with our minds that computers cannot (or couldn’t always) is not proof that we cannot be simulated by a computer. In support, I would loosely reference the Time Hierarchy theorem and a twisted form of the Halting Problem: If a Turing machine can compute more things given more time, is there always a point in your observation of such a TM where you may accurately state a problem your mind can solve that the TM cannot? Well, you can never finish knowing what the TM can compute in more time, can you?