Dienstag, 26. Juli 2011
Today, we are going on a slight diversion from the course of this blog so far, in order for me to write down some thoughts on the nature of human consciousness that have been rattling around in my head.
In a way, this is very apropos to the overarching theme of computationalism (which I personally take to be the stance that all of reality can be explained in computable terms, a subset of physicalism) that has pervaded the posts so far (and will continue to do so), because the idea that consciousness can't be reduced to 'mere computation' is often central to supposed rebuttals.
In another way, though, consciousness is far too high-level a property to properly concern ourselves with right now; nevertheless, I wanted to write these things down, in part just to clear my head.
My thoughts on consciousness basically echo those of the American philosopher Daniel Dennett, as laid out in his seminal work Consciousness Explained. However, while what Dennett laid out should perhaps most appropriately be called a theory of mental content (called the Multiple Drafts Model), I will in this (comparatively...) short posting merely attempt to answer one question, which, however, seems to me the defining one: How does subjective experience arise from non-subjective fundamental processes (neuron firings, etc.)? How can the impression of having a point of view -- of being something, someone with a point of view -- come about?
Sonntag, 17. Juli 2011
Eugene Wigner was a Hungarian American physicist and mathematician, who played a pivotal role in recognizing and cementing the role of symmetries in quantum physics. However, this is not the role in which we meet him today.
Rather, I want to talk about an essay he wrote, probably his most well-known and influential work outside of more technical physics publications. The essay bears the title The Unreasonable Effectiveness of Mathematics in the Natural Sciences , and it is a brilliant musing on the way we come to understand, model, and even predict the behaviour of physical systems using the language of mathematics, and the fundamental mystery that lies in the (apparently) singular appropriateness of that language.
Wigner's wonder is two-pronged: one, mathematics developed in one particular context often turns out to have applications in conceptually far removed areas -- he provides the example of π (or τ if you're hip), the ratio of a circle's circumference to its diameter, popping up in unexpected places, such as a statistical analysis of population trends, which seems to have little to do with circles; two, given that there is this odd 'popping up' of concepts originally alien to a certain context in the theory supposedly explaining that very context, how can we know that there is not some other, equally valid (i.e. equally powerful as an explanation) theory, making use of completely different concepts?
Dienstag, 12. Juli 2011
I have ended the previous post with the encouraging observation that if the universe is computable, then it should be in principle possible for human minds to understand it -- the reasoning essentially being that each universal system can emulate any other. But the question now presents itself: is the universe actually computable?
At first sight, there does not seem any necessity for it to be -- after all, computation and computational universality may be nothing but human-derived concepts, without importance for the universe as it 'really is'. However, we know that it must be at least computationally universal, as universal systems can indeed be build (you're sitting in front of one right now) -- the universe can 'emulate' universal systems, and thus, must be universal itself (here, I am using the term universal in the somewhat loose sense of 'able to perform every calculation that can be performed by a universal Turing machine, if given access to unlimited resources'). Thus, the only possibility would be that the universe might be more than universal, i.e. that the notion of computation does not suffice to exhaust its phenomenology.
And indeed, it is probably the more widespread notion at present that the universe contains entities that do not fall within the realm of the computable. The discussion is sometimes framed (a little naively, in my opinion), as the FQXi did recently in its annual essay contest, in the form of the question: "Is reality digital or analog?"
Freitag, 8. Juli 2011
In the mid-1930s, English mathematician Alan Turing concerned himself with the question: Can mathematical reasoning be subsumed by a mechanical process? In other words, is it possible to build a device which, if any human mathematician can carry out some computation, can carry out that computation as well?
To this end, he proposed the concept of the automated or a-machine, now known more widely as the Turing machine. A Turing machine is an abstract device consisting of an infinite tape partitioned into distinct cells, and a read/write head. On the tape, certain symbols may be stored, which the head may read, erase, or write. The last symbol read is called the scanned symbol; it determines (at least partially) the machine's behaviour. The tape can be moved back and forth through the machine.
It is at first not obvious that any interesting mathematics at all can be carried out by such a simplistic device. However, it can be shown that at least all mathematics that can be carried out using symbol manipulation can be carried out by a Turing machine. Here, by 'symbol manipulation' I mean roughly the following: a mathematical problem is presented as some string of symbols, like (a + b)2. Now, one can invoke certain rules to act on these symbols, transform the string into a new one; it is important to realize that the meaning of the symbols does not play any role at all.
One rule, invoked by the superscript 2, might be that anything that is 'under' it can be rewritten as follows: x2 = x·x, where the symbol '=' just means 'can be replaced by'; another rule says that anything within brackets is to be regarded as a single entity. This allows to rewrite the original string in the form (a + b)·(a + b), proving the identity (a + b)2 = (a + b)·(a + b).
You can see where this goes: a new rule pertaining to products of brackets -- or, on the symbol level, to things written within '(' and ')', separated by '·' -- comes into effect, allowing a re-write to a·a + b·a + a·b + b·b, then rules saying that 'x·y = y·x', 'x + x = 2·x', and the first rule (x2 = x·x) applied in reverse allow to rewrite to a2 + 2·a·b + b2, proving finally the identity (a + b)2 = a2 + 2·a·b + b2, known far and wide as the first binomial formula.
Montag, 4. Juli 2011
The way I have introduced it, information is carried by distinguishing properties, i.e. properties that enable you to tell one thing from another. Thus, whenever you have two things you can tell apart by one characteristic, you can use this difference to represent one bit of information. Consequently, objects different in more than one way can be used to represent correspondingly more information. Think spheres that can be red, blue, green, big, small, smooth, coarse, heavy, light, and so on. One can in this way define a set of properties for any given object, the complete list of which determines the object uniquely. And similar to how messages can be viewed as a question-answering game (see the previous post), this list of properties, and hence, an object's identity, can be, too. Again, think of the game 'twenty questions'.
Consider drawing up a list of possible properties an object can have, and marking each with 1 or 0 -- yes or no -- depending on whether or not the object actually has it. This defines the two sides of a code -- on one side, a set of properties, the characterisation of an object; on the other side, a bit string representing information this object contains. (I should point out, however, that in principle a bit string is not any more related to the abstract notion of information than the list of properties is; in other words, it's wrong to think of something like '11001001' as 'being' information -- rather, it represents information, and since one side of a code represents the other, so does the list of properties, or any entry on it.)
Freitag, 1. Juli 2011
Picture a world in which there are only two things, and they're both identical -- let's say two uniform spheres of the same size and color, with no other distinguishable properties.
Now, ask yourself: How do you know there are two of them? (Apart from me telling you there are, that is.)
Most people will probably answer that they can just count the spheres, or perhaps that there's one 'over there', while the other's 'right here' -- but that already depends on the introduction of extra structure, something that allows you to say: "This is sphere number 1, while that is sphere number 2". Spatial separation, or the notion of position, is such extra structure: each sphere, additionally to being of some size and color, now also has a definite position -- a new property. But we said previously that the spheres don't have any properties additionally to size and color. So, obeying this, can you tell how many spheres there are?
The answer is, somewhat surprisingly, that you can't. In fact, you can't even distinguish between universes in which there is only one sphere, two identical ones, three identical ones etc. There is no fact of the matter differentiating between the cases where there are one, two, three, etc. spheres -- all identical spheres are thus essentially one and the same sphere.
This is what Leibniz (him again!) calls the identity of indiscernibles: whenever two objects hold all the same properties, they are in fact the same object.
Now consider the same two-sphere universe, but one sphere has been painted black. Suddenly, the task of determining how many spheres there are becomes trivial! There's two: the one that's been painted black, and the one that hasn't. But how has this simple trick upgraded the solution of this problem from impossible to child's play?