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What is Intelligence?

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In a previous UD discussion I started about incompleteness I made the following affirmation: intelligence and life are not computable. A commenter kindly asked me to provide justifications for my claim. Since at UD usually I try to separate different topics in different discussions, to be more focused and reader-friendly as possible, so here is my answer in a dedicated thread. My answer unavoidably implies to investigate first what intelligence is then what life is (given the latter is an effect and the former is its cause).

A premise: intelligent design theory (IDT) per se doesn’t deal with the deep nature of intelligence or the designer. For what matters in IDT intelligence and designer can be considered as source of information. All basic results of this theory hold true when in its statements we substitute “intelligence” or “designer” or “intelligent cause” with “source of information”. This makes sense because the job of IDT is limited to investigate the signs or outputs (CSI, IC …) evidencing the inputs provided by information sources. In a sense IDT focuses on effects rather than the ultimate meaning of their cause. Despite that here I will try briefly to address something of the nature of intelligence, to satisfy the commenter’s request of explanations.

Countless definitions of intelligence were provided by philosophers and scientists according to different points of view. This very fact is sign that intelligence is complex, multi-faced and controversial topic. It’s likely any of those definitions contains some truth. However among them the pragmatic definitions cover its lower aspects only. In fact to consider intelligence as a mere tool to solve practical problems means to limit the power of intelligence to the material world and lower its ontological status to modest dimensions. We will see below that quite different appears the rank of intelligence when the problem of knowledge in its highest sense is considered. On the ground of narrow pragmatism even there may be no particular controversy between an IDer and a Darwinist. An IDer can well agree with the definitions provided by evolutionists, for example, this one by Stephen Jay Gould (“The Mismeasure of Man”): “Intelligence is the ability to face [and solve] problems in an unprogrammed creative manner”. Gould also rightly added that intelligence cannot be adequately measured (causing the ire of many psychologists). Gould’s remarks about intelligence’s unprogrammability and unmeasurability might also be in relation to the previous UD post I referenced at the beginning and in a sense agree with the thesis I am going to defend here, given the relations between the concepts of measure and computation.

Before to examine a couple of definitions of intelligence I will discuss here I must clarify what I mean for computation in this context: a deterministic finite series of instructions or operations sequentially applied to a finite set of objects. Given this definition, a computation is a mechanistic process that a machine can work out. In computability theory the archetype of such machine is the so-called Turing machine (TM).

Intelligence as generator of what is incomputable

IDT shows that CSI cannot be generated by chance and necessity (randomness and laws). An algorithm (which is a generalization of law) can output only what is computable and CSI is not. The concept of intelligence as “generator of CSI” can be generalized as “generator of what is incomputable”. Obviously, needless to say, intelligence eventually can generate also what is computable (in fact what can do more can do less). Intelligence can work as a machine but a machine cannot work as intelligence. Between the two there is a non invertible relation. This is the reason why intelligence designs machines and the inverse is impossible. To consider intelligence as “generator of what is incomputable” makes sense because we know that intelligence is able for instance to develop math. Metamathematics (Gödel theorems) states that math is in general incomputable. It establishes limits to the mechanistic deducibility but doesn’t establish limits to the intelligence and creativity of mathematicians.

Now it’s straightforward to see that the generator of what is incomputable is incomputable. Let’s hypothesize that it is computable, i.e. can be generated by a TM. If this TM can generate it and in turn it can generate what is incomputable then, given that an output of an output is an output, this TM could compute what is incomputable and this is a contradiction. Since we get a contradiction the premise is untrue, then intelligence is not computable.

The Infinite Information Source (IIS)

Now let’s pass to another more demanding but more deep perspective on our topic: intelligence as interface or link between any intelligent being and what we could call the “Infinite Information Source”. IIS is an aspect of the Metaphysical Infinity (or Total Possibility) that contains all and then contains all information too. Outside IIS there is no information because there is nothing at all. The existence of the IIS is a logical inference. In fact it is common evidence that intelligent beings (humans) routinely produce new information. This production is not creation from nothingness because from nothingness nothing comes, then this information must come from a higher source than the intelligent beings themselves. In a sense never there is new information. Besides we know from our repeated experience that intelligent beings share common information (in two senses: as information they contain inside themselves and as information they know). This proves that intelligent beings share the same higher source of information.

It remains to show that this higher source (say it S) is the IIS. The demonstration is for absurdum:

(1) Let’s hypothesize that S is finite. With “finite” I mean non Infinite (i.e. “non containing all information”). As such S is different from IIS.
(2) Since S is finite let’s consider its complement set ~S containing all information not belonging to S. Obviously ~S is included into IIS.
(3) S and ~S are disjoint sets for definition.
(4) Now consider an information ‘a’ of S and an information ‘b’ of ~S.
(5) If a and b are information, also c = (a AND b) is information.
(6) The question is: c belongs to S or ~S? It cannot belong to both because they are disjoint sets.
(7) Let’s hypothesize c belongs to S. Then S contains ‘b’, contrary to #4. Then this hypothesis is untrue.
(8) From #7 we have that c must belong to ~S. Then ~S contains ‘a’, contrary to #4. Also this hypothesis is untrue.
(9) Since we have obtained a contradiction the premise #1 is false. S is IIS.

At this point we have three basic elements in the scenario: the IIS, the being and what connects them (the channel through which information passes from the former to the latter, like a stream from a source to a sink). A classic symbolism that can help to understand their relation is the Sun that creates an image on the surface of water. The Sun is the IIS, the image is the intelligent living being and the beam connecting the Sun to its image is the channel (over-individual intellect). As the Sun is the cause of its image on water (which wouldn’t exist without it) the IIS is the cause of the intelligent living being. In particular, the intersection of the beam with the plane of our layer of existence, causes at the center of the human state the arise of human soul or psyche (with all its faculties: mind, reason, consciousness, thought, free will, emotions, sentiments …). The intersections of the beam with the center of other layers of existence cause different faculties of knowledge to other non human beings. The vertical hierarchical stack of all parallel planes represents symbolically all multiple states of being. The physical body is only the last by-product, the final unproductive production in the causality chain from IIS to matter. Warning: here the Sun is only a symbol for the knowledge’s source (traditionally light was always symbol of knowledge); obviously intelligence doesn’t really come from the physical Sun and soul is not a reverberation upon physical water. I say this because in a previous discussion about thermodynamics I defended the obvious position that the Sun does not send us information, rather energy only.

IIS is eminently incomputable because it is even un-derivable from any system (and all what comes from the development of its potentialities). In fact any system F leaves outside all what is “non F”. IIS leaves outside nothing then IIS is in principle absolutely unachievable by any systematization. Continuing the Sun’s symbolism, as the beam’s light is not really different from the source’s light, so also intelligence participates of the incomputability of IIS.

The above proof evidences also another only-seemingly odd thing: the IIS is not properly composed of parts because when we, for hypothesis, divide it into parts we obtain contradictions. It is our analytic reason that divides IIS in parts, which really don’t exist distinctively in IIS because it is eminently synthetic. IIS is essentially indivisible, and this necessarily excludes any composition and entails the absolute impossibility to be conceived as composed of parts. IIS is an aspect of the Absolute and the Absolute cannot have relations whatsoever with the relative. Since IIS really has no parts, also the link and the linked being are only apparently its “parts” and at the very end are the IIS itself. As such they directly participate of the incomputability of IIS. Again we have got the same deduction.

The same conclusion is got from yet another point of view. Let’s suppose that we find a finite process outputting intelligence. At this point nobody can a priori avoid or exclude that, through its link to IIS, intelligence receives some data that the finite process is unable to output. One can express this situation by saying that intelligence is “open” to Infinity, while, to be computable, a thing must be “closed”. Its “opening” makes intelligence virtually infinite. This is only another way to state the fundamental principle of “universal intelligibility” that sounds: there is nothing of really unknowable, all things are in principle knowable. Of course there may be countless things actually unknown to an intelligent being. But this is only a de facto temporary situation not an in principio definitive destiny. Thus we see that, as I noted above, intelligence is something far more powerful and higher than a simple tool for solving practical problems, because virtually can know all. Since intelligence is virtually the knower of all what is incomputable, in turn it cannot be computable because the knower cannot be lower in rank than the known.

Given we are dealing with universal intelligibility it is necessary to clear a possible misunderstanding. To avoid it we must carefully distinguish reason and intellect. This distinction, which was well clear to most ancient philosophers, was lost at the arise of rationalism and humanism in the modern era. As someone said: “it was reason to betray intellect”. The first product of rationalism in the scientific field was Cartesian mechanicism, which is in relation with computability I deny here when applied to intelligence and life. Reason is merely an individual human faculty. It is a discursive indirect form of analytical knowledge that takes as support logic and argumentative tools. Reason cannot be universalized as is. Quite differently intellect is a higher universal faculty of direct synthetic knowledge pertaining to all states of being. This explains because with the arise of rationalism and humanism the knowledge of universal principles (as the Metaphysical Infinity) was lost: what is universal can be known only by a universal faculty. Reason is only the lower individual part of intelligence (the horizontal image), while intellect is its higher over-individual part (the vertical beam). Intellect is over-rational. Warning: over-rational is not at all irrational as some believe! The universal intelligibility makes sense only when addressed by intellect. If we remain on the plane of human reason, there is no universal intelligibility. In other words it is not reason to be omniscient and there is no such thing as universal reasonability.

The key point to focus is that all the above definitions of intelligence agree and support each other. They are consistent because represent different viewpoints of the same reality. Hence also the respective demonstrations of incomputability show the same impossibility seen from different perspectives. The above argument has corollaries. The incomputability of intelligence and its non mechanistic nature debunks once and for all any illusion of the so-called Artificial Intelligence to create real intelligence. The IIS can be considered an aspect (expressed in term of information) of the Universal Intelligence or Divine Intellect and since it is also the Source of the universe, which is a design, the symbolism of the Great Designer can be applied to it.

Life as carrier of intelligence

Now let’s consider life (specifically life of conscious living beings) and give of it the following definition: the physical carrier or support of intelligence, what allows intelligence to manifest and operate on the physical plane. If the carrier (living soul and body) were mechanistic only they could not adequately express intelligence, which is not mechanistic. It is a claim of IDT that physical signs manifest the non physical nature of intelligence. These signs (CSI, IC, etc.) are non mechanistic and what displays such signs cannot be mechanistic too. Living soul and body display such signs and then we can conclude that life is non mechanistic.

To illustrate with an example the concept, let’s consider a clear manifestation of intelligence in a living being: language. Also Noam Chomsky admits that language is structural and hardwired in its physical carrier, the brain. Language is not mechanistic: the high expressions of literature cannot be created by a machine. The classic objection of materialists to this claim is: also machines can output literature works. Machines can output texts (strings of characters), but their outputs fully lack meaning and indeed this proves that they are not true manifestation of intelligence (which is the only source of meanings). For instance, when a writer writes the four-chars word “love” he has in mind all the meanings of the idea, instead when a machine writes “love” it has nothing in mind for the simple fact that it has no mind. And here what stays in the “background” (the semantic) is more important and essential of what stays in the “foreground” (the syntax), so to speak. Moreover if a machine writes “love” it is because was programmed to do so, not because it wanted that (as a human writer does). Just a curiosity: an ancient Hebrew legend speaks of the Golem, a sort of automaton that they say Cabalists were able to vitalize by mean of esoteric rituals. The Golem was able to simulate a living being (a robot ante litteram) but it wasn’t able to speak because language is an advanced ability that only real intelligent living beings have.

Of course all that I have written here is light-years from the materialist and reductionist Darwin’s idea of “thought, being a secretion of the brain”. Modern evolutionists believe to be more sophisticated saying that “thought is emergent property of the brain”. But if we examine it their claim is not more explicative. In fact emergent properties involving information (and mind eminently implies information) don’t spontaneously “emerge” from the bottom like a secretion (as they think) but come from the top, from an intelligent source. About this topic see my previous post on emergence.

To sum up about intelligence (like many other things) we are before two diametrically opposite worldviews: the ID non materialist and the materialist (with all its consequence, evolutionism included). The former is a top-down worldview while the latter is a bottom-up approach. Non materialism states that matter itself comes from information. Materialism, at the very end, denying any higher principle than matter, believes that information arose from matter. These two opposite worldviews cannot be both true. I hope these brief notes may help some to know which of them is on the side of truth.

Comments
CJYman at 96, Thanks for the detailed response. I still have some significant issues with "specification", but I'll hold those for now and focus on what I believe are two key problems with your formulation of CSI. CJYman: “then we can calculate for specificity, and then calculate the probability of the event given a uniform probability distribution.” Mustela: “Why a uniform probability distribution?” Because chance (randomness) is characterized statistically as a uniform probability distribution and all that CSI is doing is removing the chance hypothesis. Mustela: “We know from observation that many mechanisms identified by modern evolutionary theory are not random in their behavior (mutation is assumed to be, but selection, for example, is not).” Of course evolutionary algorithms aren’t random. That demonstrates that a uniform probability distribution is not biologically relevant. If you are using this calculation to demonstrate that biological evolution by known natural processes cannot account for the diversity of life that we observe then you must take into account those known natural processes. Once CSI is determined to be “present,” then we know that we aren’t dealing with a uniform probability distribution. I'm sorry, I don't follow that. I thought we were calculating CSI here. To do so, we need to take into account everything we know about the biological artifact under consideration. However, merely having a non-uniform probability is still no help. According to the NFLT, we need a matching between the non-uniform distribution and the correct search algorithm which can make use of that distribution. That’s how we get an evolutionary algorithm. The No Free Lunch theorems merely state that any particular search strategy is only as good as random search over the space of all possible fitness landscapes. We're not dealing with a space of fitness landscapes, we're dealing with the physical reality we all know. That's one fitness landscape and the NFL theorems say nothing about the ability of evolutionary mechanisms to perform better than random search. Mustala: “The uniform probability distribution assumption is equivalent to the assumption that the DNA arose in its present form de novo. That does not correspond to either modern evolutionary theory nor to observed evolutionary mechanisms.” Exactly, since CSI really only tells us that chance is not a viable option. Therefore the only two options left on the table are intervention by an intelligence or an evolutionary algorithm. Now, the reason CSI is used to signify previous intelligence is because ID proponents claim that the fortuitous matching between search algorithm and non-uniform distribution is just as difficult to arrive at by chance as is the original pattern in which CSI was measured. So if chance can’t get us to the original pattern, it won’t get us to the evolutionary algorithm. In fact, I see evolutionary programming as a hallmark of intelligence. There are a lot of assumptions in this one paragraph. First, we haven't seen an actual calculation of CSI, so claiming that CSI shows anything is premature. Second, there are more choices than an intelligent agent or an evolutionary algorithm. Known physical and chemical processes can't be ruled out. Third, there is no reason to consider the search algorithm to be "matched" to anything. We live in a universe with a particular set of physical laws (for lack of a better term). Those are a given. There is no "search for a search" taking place. Mustela Nivalis: “the mechanisms identified by modern evolutionary theory do not create things like genomes de novo — they do so incrementally and therefore cannot be modeled so naively.” CJYman: “The point is that according to Marks and Dembski’s recent work on active information, it is just as improbable to find something such as CSI as it is to generate a set of laws that would create an evolutionary algorithm to produce an incremental pathway to that CSI in question.” Mustela: “But we don’t need to find a set of laws. Physics, and therefore chemistry, is a given. Modern evolutionary theory attempts to explain what we observe, with the physical laws that exists.” Physicists are also interested in fine tuning of laws to produce life and the ratios of variables of those laws which would allow life to exist and evolve vs. which variables would not even allow life enough time or material or non-uniformity to form. BTW, I am in no way an “interventionist.” I'm afraid that doesn't address the problem with your calculation. There is no "search for a search". The NFL theorems don't apply. Known evolutionary mechanisms work far better than random search in this universe. Mustela: “You might be able to construct an argument for cosmological ID based on “search for a search”, but if you want to apply CSI to real biological systems, you have to take known evolutionary mechanisms into account.” If what I have explained above is true, then any measure of CSI will show that chance is not capable of producing said pattern/event, evolutionary algorithm or not. Because you fail to take into consideration the physics of this universe and the known mechanisms of modern evolutionary theory, your calculation has no biological relevance. Can you modify your calculation to address these issues? I'm still very interested in implementing a CSI calculator in software.Mustela Nivalis
December 17, 2009
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CJYman: "First, according to Dr. Dembski, a specified event is an event which can be formulated as an independent pattern." Mustala: "I’ve read that, but it seems very difficult to apply. What does it mean for a pattern to be independent? What is an example of a dependent pattern? Can’t anything be described by a pattern separate from itself?" I've explained "specified patterns" in more detail in this thread on my blog and in the thread "Polanyi and Ontogonetic Emergence" comment #8 here on UD. Sorry 'bout the lack of link. I couldn't get it to work for some reason. Now, I'll let you answer your own question of "can't anything [be specified]." Here are two events: 1. You look on your computer and you see a string of symbols "Hello Mustela, how are you today?" The meaningful event that has appeared on your screen is specified since it conforms to a pre-set pattern of rules for the english language and the english dictionary (including slang). So f(english language+dictionary)= meaningful sentences -- when operated by a semiotic agent whether that be a computer program or human. That is why a semiotic agent is important -- the agent is the one that specifies the pattern to the event. 2. You wake up early in the morning and you go to the mailbox to get your mail. You notice that the weather is getting colder and leaves are beginning to fall. You see an event of leaves scattered on your front lawn. Is this event specified? Mustela: "Are you using this mathematical notation formally or informally? If formally, could you explain the nature of the function f?" I am using this mathematical notation because in f(x)=y, x is independent of y. Therefore, f(pattern)=event, I believe is a formal way to show the specified event/pattern relationship. The given event is literally a function of a pattern. Therefore, the event is specified. CJYman: "So, if we have an event such as a folding and biologically useful protein," Mustella: "A protein is an event? I apologize if this is a simple question, but I have read a fair bit of ID material and I find this terminology confusing. Can you rephrase it in more standard biological terms?" I don't think there is a need to get overly technical with the word "event" other than to use it as "something that happens" -- basic dictionary definition. Yes, a pattern of amino acids and ultimately a pattern of DNA can produce a folding and biologically useful event -- a protein. CJYman: "and an independent pattern such as a stretch of DNA," Mustela: "Again, I apologize if I appear pedantic, but why would we consider the DNA that encodes for a protein to be an “independent pattern”? What is the precise definition you’re using?" No problem, I appreciate when people ask for clarification unless the clarification descends into continual nitpicking. Independent here is used as independent is used in '"x" is independent of "y" in f(x)=y.' Again, that is why I believe that if you follow a function where f(patter)=event, you can't go wrong. CJYman: "then we can calculate for specificity, and then calculate the probability of the event given a uniform probability distribution." Mustela: "Why a uniform probability distribution?" Because chance (randomness) is characterized statistically as a uniform probability distribution and all that CSI is doing is removing the chance hypothesis. Mustela: "We know from observation that many mechanisms identified by modern evolutionary theory are not random in their behavior (mutation is assumed to be, but selection, for example, is not)." Of course evolutionary algorithms aren't random. Once CSI is determined to be "present," then we know that we aren't dealing with a uniform probability distribution. However, merely having a non-uniform probability is still no help. According to the NFLT, we need a matching between the non-uniform distribution and the correct search algorithm which can make use of that distribution. That's how we get an evolutionary algorithm. Mustala: "The uniform probability distribution assumption is equivalent to the assumption that the DNA arose in its present form de novo. That does not correspond to either modern evolutionary theory nor to observed evolutionary mechanisms." Exactly, since CSI really only tells us that chance is not a viable option. Therefore the only two options left on the table are intervention by an intelligence or an evolutionary algorithm. Now, the reason CSI is used to signify previous intelligence is because ID proponents claim that the fortuitous matching between search algorithm and non-uniform distribution is just as difficult to arrive at by chance as is the original pattern in which CSI was measured. So if chance can't get us to the original pattern, it won't get us to the evolutionary algorithm. In fact, I see evolutionary programming as a hallmark of intelligence. Mustela Nivalis: “the mechanisms identified by modern evolutionary theory do not create things like genomes de novo — they do so incrementally and therefore cannot be modeled so naively.” CJYman: "The point is that according to Marks and Dembski’s recent work on active information, it is just as improbable to find something such as CSI as it is to generate a set of laws that would create an evolutionary algorithm to produce an incremental pathway to that CSI in question." Mustela: "But we don’t need to find a set of laws. Physics, and therefore chemistry, is a given. Modern evolutionary theory attempts to explain what we observe, with the physical laws that exists." Physicists are also interested in fine tuning of laws to produce life and the ratios of variables of those laws which would allow life to exist and evolve vs. which variables would not even allow life enough time or material or non-uniformity to form. BTW, I am in no way an "interventionist." Mustela: "You might be able to construct an argument for cosmological ID based on “search for a search”, but if you want to apply CSI to real biological systems, you have to take known evolutionary mechanisms into account." If what I have explained above is true, then any measure of CSI will show that chance is not capable of producing said pattern/event, evolutionary algorithm or not. Oh, and referencing any questions about semiotic agents, a semiotic agent is merely an agent that utilizes sign systems, therefore does not have to be either human or conscious. The cell itself is a semiotic agent and thus we use the language of the cell (the language of life) to specify and measure for the CSI of a biological object. In fact, we don't even have to specify anything, the cell has already done that by objectively converting patterns into events.CJYman
December 17, 2009
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... all I have time to say right now, is that the data for any equation is affected by human ignorance and gaps in our knowledge. Garbage in, garbage out. This is what I have been trying to explain to Zachriel in "The Odds That End: ...." thread. Yet this doesn't effect bringing the age and size of the universe into scientific territory, so for that reason, a measurement of CSI is definitely in scientific territory. CSI, like any equation (such as those used to calculate the age and size of the universe) relies on estimations of presently available data. And, as already explained, I gave the extreme benefit of the doubt to the skeptic in my assumptions ... so yes, the measurement of CSI could easily be updated in either direction in the future. The question is: "what direction will be the overall trend" or will we see the measurement(s) over time and given the error values of each variable stabilizing around an average value? With everything that we know can we even attempt to get a non-CSI value for the protein Titin? If someone can go back to my measurement and find the -log 2 of the end measurement, that would be great so that we can see the size of the value for CSI that we are presently dealing with. And then, how far up the hierarchy of protein multiple protein complexes can we go until there is no reasonable way to even fudge data to get a non-CSI value? Excellent questions for further ID research. In fact, I offer the invitation to the ID critics to help out with this research. I have provided an explanation of how to calculate for CSI above. Mustela and ROb, I am enjoying reading through your questions and comments and I hope that I have time soon to get back to you guys.CJYman
December 17, 2009
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Hello Zach, Mustela, and ROb. Sorry I'm late to rejoin the discussion. I was following this thread on another computer and then couldn't remember in which thread I was having this discussion. I would love to continue, but am getting too busy again. We will definitely have to continue our discussion of CSI in the future. I now have the relevant threads bookmarked. ...later...CJYman
December 17, 2009
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R0b #92 Sorry for the delayed response (I just returned from a 5-day extended weekend). What was my aim in writing my paradox? It was to indicate that the total information can be fractioned only apparently and illusorily. Even I stated it explicitly in the article as follows:
The above proof evidences also another only-seemingly odd thing: the IIS is not properly composed of parts because when we, for hypothesis, divide it into parts we obtain contradictions.
Therefore when you write:
The real root of your absurdum is the following: A set of logical statements cannot be partitioned into subsets each of which are closed under logical derivation. So in your case, the premises that S and ~S are closed under logical derivation result in a contradiction.
you say almost the same thing from the point of view of logic and I agree with your statement for the simple fact that this contradiction was indeed what I wanted to obtain and emphasize. Just before writing step #1 I knew in advance that it was a flawed premise and a mere impossibility. What are for absurdum arguments but arguments where one states an untrue (or alleged untrue) premise to see the consequences? After all, my premises were a unique premise (although expressed in some steps): A and ~A are disjoint. As a consequence when we obtain the contradiction it is this premise to fall, what else. This premise outputs a contradiction because such sets cannot be disjoint, in that a simple logical AND upon any two elements of them immediately shows us that they cannot be close (this is the job of steps #5-9). Hence, why do you write "your conclusion is a non sequitur" when my conclusion is to refute the premise of impossible closure contained in step #1 exactly as you do? It seems to me there is agreement between us on this ground. In one of its interpretations, my article wanted to underline that finite information sources cannot be considered really separated from a limitless information source (this is another aspect related to the above conclusion). I could have written it very synthetically as "Total Truth is indivisible" or "Total Possibility has no parts" (both axiomatic in metaphysics) or something like that but I was afraid the thing passed almost unobserved and somehow abstract. So I thought that a 9-step analytic paradox expressed in terms of set theory and showing the unity of logic (which represents sort of "image" or "symbol" of that metaphysical Unity) might generate more interest and debate. Indeed the many technical comments of you and JT show the idea was good. I admit that my paradox can be misunderstood depending on the interpretations one can assume about. Different interpretations are due to the fact that it is far from being rigorously formalized according to mathematical logic. I never pretended to have provided such exactness of formalization and of course I thank whoever helps to deepen and improve it somehow (as you and JT cleverly did). My goal was only to offer material to the reader for meditating about what, expressed in few words, is the infiniteness and indivisibility of information.niwrad
December 8, 2009
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niwrad, I hope you had an enjoyable vacation.
Also if S contained “all statements logically derivable from those it contains” (as you added to refute my argument) my proof would remain valid.
Reductio ad absurdum proves that the premises cannot all be true simultaneously. It doesn't show that any given premise is false, so your conclusion is a non sequitur, and adding more premises only exacerbates the problem. The real root of your absurdum is the following: A set of logical statements cannot be partitioned into subsets each of which are closed under logical derivation. So in your case, the premises that S and ~S are closed under logical derivation result in a contradiction. This is proven easily by noting that "a OR not a" follows from any set of axioms, but it cannot reside in multiple disjoint sets.R0b
December 5, 2009
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JT #87-90 Unfortunately it seems we never will converge to an agreement. No problem. What is important is that anyone feels comfortable with his own ideas. Anyway I agree with you that the best thing is to stop here our diatribe to not bother the readers (also if I would like to continue the discussion with you (and R0b too)). You are a fair and acute debater and I thank you for your contribute to the discussion. Good weekend. P.S. I am not from the U.S. and, last but not least, today I am in vacation.niwrad
December 4, 2009
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Also if S contained “all statements logically derivable from those it contains” (as you added to refute my argument) my proof would remain valid. What matters for its validity is that S contains a but doesn’t contain b and that ~S contains b but doesn’t contain a. And (once again) if S contains 'a AND b' that is not the same as containing 'b', as 'b' and 'a AND b' are not equivalent. (And it was not my original intention to belabor this obvious point to such an extent.)JT
December 3, 2009
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Now I'm getting confused - I'm not sure if I characterized your argument at the top of 87 correctly. Oh well, if my orginal point in 63 still isn't clear, so be it.JT
December 3, 2009
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w/ apologies for the personal tone.JT
December 3, 2009
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Also if S contained “all statements logically derivable from those it contains” (as you added to refute my argument) my proof would remain valid. What matters for its validity is that S contains a but doesn’t contain b and that ~S contains b but doesn’t contain a.
No, no you're completely confused. My argument wasn't dependent at all on how I characterized S. AS R0b pointed out to you, if S is not as I described, then c can be in S, because c in S in that case does not mean b is in S. The quote Bill Dembski provided earlier today bears repeating: “I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.” Leo Tolstoy A question that comes to my mind is, what does etiquette dictate a person do in a forum such as this when he has clearly been proven wrong. Interesting, what is Twilight Zone? Wow - you really are young aren't you (or not from the U.S.) There were two T.V. series over twenty years apart (one in the mid 80's) with that name.JT
December 3, 2009
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JT #82-84
If S does not necessarily contain all statements logically derivable from those it contains then it need not contain a or b merely by virtue of containing c.
Also if S contained "all statements logically derivable from those it contains" (as you added to refute my argument) my proof would remain valid. What matters for its validity is that S contains a but doesn’t contain b and that ~S contains b but doesn’t contain a.
The fact is, the reason I was able to refute this is I believe I’ve seen it somewhere before.
Your refutation is flawed due to what I wrote above.
But that bit about a computer not being able to understand the word “Love” I believe was in an old Twilight Zone episode.
Interesting, what is Twilight Zone?niwrad
December 3, 2009
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niwrad, part of the problem is the phrase "a is true in S", which seems to combine the propositions "a is true" and "a is in S". If you avoid the combined phrase and keep the propositions distinct, I think there will be a lot less confusion.R0b
December 3, 2009
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But that bit about a computer not being able to understand the word "Love" I believe was in an old Twilight Zone episode.JT
December 3, 2009
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correction: If S does not necessarily contain all statements logically derivable from those it contains then it need not contain a or b merely by virtue of containing c.JT
December 3, 2009
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If c is true in S (then a is true in S and b is true in S) I don’t understand why you say that a is not true in S. Hence #7 is not a non sequitur as you say.
could be restated as:
If c is in S (then a is logically derivable from the statements in S and b is logically derivable from the statements in S) I don’t understand why you say that a is not logically derivable from the statements in S.
No one said that. You previously denied my defintion of S as containing all statements logically derivable from those it contains. We know its an incomplete set anyway, because that's how you defined it to begin with. If S does not necesarily contain all statements logically derivable from those it contains then it need not contain a or b merely by virtue of containing a. The fact is, the reason I was able to refute this is I believe I've seen it somewhere before. So you might credit wherever it is you got it from if you know.JT
December 3, 2009
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R0b #76, JT #77 I answer both because your replays are conceptually the same.
If S is not implicitly defined as JT says, then (7) contains a non sequitur. It does not follow from “(a AND b) ? S” that “a ? S”.
If c is true in S (then a is true in S and b is true in S) I don’t understand why you say that a is not true in S. Hence #7 is not a non sequitur as you say.
Pick your poison. Your proof ostensibly shows that there is no such thing as a finite set of information. Do you really believe that this conclusion can be reached through valid logic?
My proof (and my article as a whole) was aimed to show that information and logic necessarily point to IIS, that an infinite information source is metaphysically and logically necessary. There is nothing more logically valid than IIS. If you admit and recognize IIS it is not important that you like or dislike my proof. My proof is only the finger pointing to the Sun. If one sees directly the Sun the finger is no longer necessary. If you do not admit and recognize the Sun all what I can do is to provide a finger pointing to it, I cannot see the Sun for you.niwrad
December 3, 2009
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niwrad: You may have misinterpreted the following
if S is some set of axiomatic statements plus all statements logically derivable from those statements, then ~S is just every other possible statement.
~S is every other possible statement by virtue of the definition of ~S, not as a result of how I have defined S (and my definition was an interpretation of your remarks in 61.) Also, I notice you say that ~S is all information not in S (thus implying you may be employing a custom defintion of ~). However you also said 'a AND b' was information, so if its not in S (which it can't be by the above definition) then it has to be in ~S (without contradiction).JT
December 3, 2009
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William, thanks for the references. It appears to me that three separate concepts are being conflated here. #1 Computability, as defined in computability theory. #2 Complexity, as defined in complexity theory. #3 Probability, as defined in probability theory. Computability theory and complexity theory are branches of computing theory, which deals with deterministic computation. Probability theory obviously deals with non-deterministic events. niwrad's stated definition of computability is the computability theoretic definition, so we would expect him to be referring to concept #1. The two papers you referenced deal with computational complexity, concept #2. CSI deals with probability, concept #3. We seem to be jumping around among three different questions, each of which is addressed by a different concept: #1 -- Can any conceivable TM generate X as an output? #2 -- Can problem P be solved in time t with memory m? #3 -- Is event E likely to occur at least once given n probabilistic events? I think we need to untangle this conflation before we can have a coherent discussion regarding niwrad's post.R0b
December 3, 2009
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(i.e. without a,b actually being in the set - correct me if I'm wrong.)JT
December 3, 2009
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niwrad: Your proof is that the statement 'a AND b' cannot be in either S or ~S because it results in a contradiction, and therefore ~S does not exist. I show that 'a AND b' can be in ~S without any such contradiction. And if S is not defined as I state (an interpretation of your response to me in 61 actually) then as R0b says 'a AND b' can be in S (and thus not in ~S by the definition of ~). I think you're still assuming that just putting the statement 'a AND b' in a set automatically gives set membership to a and b individually as well.JT
December 3, 2009
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niwrad:
“S is some set of axiomatic statements plus all statements logically derivable” – this is not my definition of S.
If S is not implicitly defined as JT says, then (7) contains a non sequitur. It does not follow from "(a AND b) ∈ S" that "a ∈ S". Pick your poison. Your proof ostensibly shows that there is no such thing as a finite set of information. Do you really believe that this conclusion can be reached through valid logic?R0b
December 3, 2009
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Interesting paper here on the computational capacity of the universe. http://www.scribd.com/doc/15664694/Computational-Capacity-of-the-Universe-WWWOLOSCIENCECOM And another one on the finite information processing bounds of the universe: http://cosmos.asu.edu/publications/papers/EmergentBiologicalPrinciplesandtheComputationalPropertiesoftheUniverse.pdf The latter paper argues that some emergent sorting process is required to explain functional biological features, because it is otherwise beyond the brute computational capacity of the universe.William J. Murray
December 3, 2009
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#73 It’s not being argued that the CSI can be respresented (i.e., exists), but rather that it cannot be computed without intelligence. As far as I know the only things that ever computed anything were people - unless you include man-made equipment designed to compute things.Mark Frank
December 3, 2009
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Rob, You're correct. I was thnking of a diffferent issue I was currently reading about, about where a sufficiently large factoring problem gets computed, such as 10^200^200 (or something like that) which is computable in shorthand, but which would take more quanta than the universe has availabe to write out. However, that doesn't change the computability problem. It's not being argued that the CSI can be respresented (i.e., exists), but rather that it cannot be computed without intelligence. Your long notation and my shorthand one were computed by intelligence, not by unintelligent chance and law. Thanks for the correction though.William J. Murray
December 3, 2009
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William:
IOW, 10^200 as an expression is computable only by intelligence, not by the physical universe. You can’t even write the number out because there aren’t enough quanta in the known universe to do so.
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000. (There are 200 zeros there, even if your browser doesn't show them all.) I just pwned the universe and violated the SLoT at the same time. All before breakfast.R0b
December 3, 2009
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niwrad, In your response to Rob about the computability of CSI, I think you and he missed a more fundamental point; the CSI of such systems are not computable by the physical universe because it lacks the computing resources; such CSI is only computable by intelligence such as ours. IOW, 10^200 as an expression is computable only by intelligence, not by the physical universe. You can't even write the number out because there aren't enough quanta in the known universe to do so. Let's remember, no one is disputing that life exists, or that DNA exists; the question is whether or not the physical universe has the capacity to compute it without intelligence.William J. Murray
December 3, 2009
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Mr Joseph, And if science is grounded in reality then what does that do to the theory of evolution which is grounded in personal biases? Oh, don't worry about that problem any more. The FSM told me that Charles Darwin was actually copying the text of his books from golden plates of pasta (alphabet, of course though some splitters think noodles forming script words - they are going to Hull to sort laundry forever, believe me!) shown to him by the Lady of the Lake, errr, Warm Little Pond. So have no fear, the theory of evolution is on extremely solid footing of direct revelation, we now know.Nakashima
December 3, 2009
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R0b #65
JT, you’ve explained niwrad’s fallacy perfectly.
Not true, see my comment #66.
His reductio ad absurdum is invalid because it contains hidden premises, namely that S and ~S are each closed under logical derivation. Eliminate the premise that ~S is closed under logical derivation, and no absurdum follows from the remaining premises, as you show.
I stated that S and ~S are disjoint in step #3, then no hidden premises.
But if the proof were valid, we could come up with all kinds of fun corollaries. If S=IIS for all finite sets S, then it would follow that IIS is simultaneously equal to every finite set, including the empty set. So it follows that information doesn’t exist, and therefore this thread doesn’t exist.
The truth is exactly the opposite. My proof, far from showing that IIS defaults "to every finite set, including the empty set", evidences the indivisible nature of information and the infiniteness of IIS. Not that my proof were necessary because IIS is a priori truth and rather it would be its non existence to be nonsense.niwrad
December 3, 2009
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And if science is grounded in reality then what does that do to the theory of evolution which is grounded in personal biases?Joseph
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