The cause of incompleteness
|November 3, 2009||Posted by niwrad under Intelligent Design|
In a previous post I promised to start at UD a discussion about the incompleteness of physics from an ID point of view. Here is the startup article.
At Aristotle’s time “physics” was the study of nature as a whole. In modern times physics seems to have a more specific meaning and is only one field among others that study nature. Nevertheless physicists (especially materialist ones) claim that physics can (or should) explain all reality. This claim is based on the gratuitous assumption that all macroscopic realities can be deduced entirely from their microscopic elements or states. Also if this assumption were true there would be the problem to understand where those fundamental objects or states came from in the first place. Many physicists even think about a “Theory of Everything” (ToE), able to explain all nature, from its lower aspects to its higher ones. If a ToE really existed a system of equations would be able to model every object and calculate every event in the cosmos, from atomic particles to intelligence. The question that many ask is: can a ToE exist in principle? If the answer is positive we could consider the cosmos as a giant system, which evolution is computable. If the answer is negative this would mean that there is a fundamental incompleteness in physics, and the cosmos cannot be considered a computable system. An additional question is: what are the relations between the above problem and ID?
Stephen Hawking in his lecture “Gödel and the end of physics” seems to think that Kurt Gödel’s incompleteness theorems in metamathematics can be a reason to doubt the existence of a ToE:
“Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel’s theorem ensured there would always be a job for mathematicians. I think M-theory will do the same for physicists. I’m sure Dirac would have approved.”
In two words, Gödel’s incompleteness theorems essentially say that in general a mathematical formal system beyond a certain complexity is either inconsistent or incomplete. Hawking’s reasoning is something like this: every physical theory is a mathematical model, and since, according to Gödel’s incompleteness theorems, there are mathematical results that cannot be proven, then there must be physical statements that cannot be proven as well, including those contained in a ToE. Gödel’s incompleteness applies to all mathematical theories with potentiality greater or equal to arithmetic. Since any mathematically described physical theory has potentiality greater than arithmetic then is necessarily incomplete. So we are before a fundamental impossibility of a complete ToE that comes from results in metamathematics.
Computability theory and its continuation Algorithmic Information Theory (AIT) are mathematical theories that can be considered sort of meta-informatics, because are able to prove statements about algorithms and their potentiality, what they can or cannot output. A basic concept of AIT is compressibility: an output that can be generated by a computer program with binary size much lesser than the output itself is called “compressible” or “reducible”. Given that a mathematical formal system and its theorems are comparable to an algorithm and its outputs we find that incompleteness in math (improvable theorems do exist in a formal system) has its equivalence in incompressibility in AIT (non algorithmable incompressible outputs do exist). For these reasons by mean of the tools of AIT it is possible to prove theorems equivalent to Gödel’s theorem. According to Gregory Chaitin (the founder of AIT):
“It is sometimes useful to think of physical systems as performing algorithms and of the entire universe as a single giant computer” (from “Metamathematics and the foundations of mathematics”). – “A theory may be viewed as a computer program for calculating observations. This provides motivation for defining the complexity of something to be the size of the simplest theory for it, in other words, the size of the smallest program for calculating it” (from “On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility”).
A physical theory is composed of laws (i.e. algorithms). If the universe is a giant computer then the incompressibility results of AIT apply: incompressible outputs do exist, which no algorithm can create, then a complete physical theory describing those outputs does not exist. If the universe is not a giant computer then a complete physical theory describing it does not exist for definition. In both cases we arrive to the incompleteness of physics. The conclusions of Chaitin are somewhat similar to Hawking’s ones:
“Does [the universe] have finite or infinite complexity? The conventional view on this held by high-energy physicists is that a ToE, a theory of everything, a finite set of laws of nature that we may someday know, which has only finite complexity. So that part is optimistic! But unfortunately in quantum mechanics there is randomness. God plays dice, and to know the results of all God’s coin tosses, infinitely many coin tosses, necessitates a theory of infinite complexity, which simply records the result of each toss!” (“From Philosophy to Program Size” 1.10)
About the infinite complexity Chaitin is correct. But the language of Chaitin is a bit misleading where he says that “God plays dice”. In reality also all apparently random results are wanted by God. Otherwise His will would be limited by dice and this is nonsense. Also randomness is under the governance of God. To deny this would mean to deny its Omnipotence, then deny the Total Possibility (which is another name for what theology calls God’s Omnipotence). From this point of view any result that appears random to us is simply an event which unique cause is directly God himself (the First Cause), while a result due to a physical law is an event wanted by the Laws-Giver too obviously, but by mean of an intermediary law (which works as secunda causa). So events divide in two sets: those wanted by God not compressible into laws and those wanted by God compressible into laws. After all there is no reason to believe that God should limit himself to the latter only.
There is another point of view from which a physical ToE is incomplete. We might call this argument, “physics-incompleteness argument from ID”. If a ToE must be indeed what it wants to be, that is a theory describing all aspects of reality, this ToE must also deal with its higher aspects, those related to intelligence. But intelligence is what creates theories. In fact a ToE is an intelligent design and physicists who develop it are designers. A ToE is incapable to compute the decisions of its designer. Said other way, the free will of the designer of a ToE entirely transcends it. You can also look at the problem from this point of view: if a physicist decides to modify the ToE, the ToE cannot account for it, because it is impossible that a thing self-modifies. As a consequence, since a ToE doesn’t compute all things in the universe, is incomplete and not at all a ToE.
To sum up we have that metamathematics proves the incompleteness of math. AIT proves the incompressibility of informatics. Both these results reverberate on physics causing its irreducibility. In turn ID shows that a ToE is incomplete because cannot compute its designer. These three fields agree to show the incompleteness of physics and compose a final consistent scenario.
The important thing to get is that all incompleteness results in specific fields are only particular cases of a more general truth. To understand it we must start from the fundamental concept of the aforesaid Total Possibility, which has no limits because leaves outside only the impossible and the absurd that are pure nothingness. For this reason, the Total Possibility is not reducible to a system. In fact any defined system S leaves outside all what is ‘non S’. This ‘non S’ limits the system S. Since S has limits it cannot be the Total Possibility, which is unlimited. As Leibniz said: “a system is true for what affirms and false for what denies”. Also large-enough sub-sets of the Total Possibility are not reducible to systems. For Gödel “large-enough” means with potentiality greater or equal to arithmetic. Mathematics and the cosmos are large-enough in this sense and as such are irreducible to systems. They are simply too rich to be compressed in a system because they are aspects or functions of the Total Possibility. The Total Possibility has nothing to do with the simple infinites (mathematical or of other different kinds). Any particular infinite has its own limits, in the sense that leaves outside something (e.g. the infinite series of numbers doesn’t contain what was before the Big-Bang, galaxies, elephants, your past and future thoughts, what will remain when the universe will collapse … while Total Possibility does). While there is only one Total Possibility there are many infinites, which are infinitesimal compared to it. To confuse the two concepts, Total Possibility and infinites, is a serious error and cause the total misunderstanding of what the former is.
Systematization (the reduction or compression to a system) represents epistemologically also all the bottom-up approaches to get the total knowledge. The fundamental failure of systematization, when applied to rich-enough sub-sets of the Total Possibility is also the failure of all bottom-up reductionist and positivist approaches to knowledge. Of course this failure appears negative only for who hosts the naive illusion that more comes from less. For who understands the Total Possibility, the failure in principle of systematization is only a logical consequence of the fact that less always comes from more.
To use a term from the computers jargon that all people understand, mathematics and the cosmos are “windows” on the Total Possibility. As a window-shell on our display is an aperture on the operating system of our computer and allows us to know something of it, analogously mathematics and the cosmos are large-enough apertures on the Total Possibility. This is sufficient to make them not systematizable. This is true also for the cosmos despite the fact it is infinitesimal respect the Total Possibility. It is easy to see that such “window” symbol is equivalent to the symbolism of “Plato’s cave”, from which the prisoners can see only the shadows of the realm of Ideas or Forms (Plato’s equivalent of the eternal possibilities contained in the Total Possibility). Plato, although he sure didn’t need scientific confirmations for his philosophy, would be glad to know that thousands years after him fundamental results in science support his correct top-down philosophical worldview.
Given its fundamental incompleteness, math implies the necessity of the intelligence of mathematicians for its endless study. Since informatics is basically incompressible, computers (and in general Artificial Intelligence) will never be able to fully substitute human intelligence. Given its fundamental incompleteness, physics implies the necessity of the intelligence of physicists for its endless development. In turn ID says exactly the same thing about complex specified information: its generation will always need intelligent designers. In a sense also the ID concept of irreducible complexity agrees with the above results: in all cases there is a “true whole” which richness cannot be reduced, indeed because it represents a principle of indivisible unity. The final victory of ID against evolutionism will be only the unavoidable consequence of the fact that the former is a correct top-down conception while the latter is a bottom-up illusion. Bottom-up doesn’t work for the simple fact that reality is an infinite hierarchy of information layers, from the Total Possibility all the way down until to the more infinitesimal part of the cosmos.
A believer asked God: “Lord, how can I approach You?” God answered: “By mean of your humility and poverty”. May be in this teaching there is a message for us about our actual topic (a message that Gödel, Hawking and Chaitin seem just to have humbly acknowledged): indeed by recognizing the radical incompleteness (“humility and poverty”) of our systems, we have a chance to understand the “Infinite Completeness” of God.