Uncommon Descent Serving The Intelligent Design Community

Human Consciousness

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(From In the Beginning … ):

For the layman, it is the last step in evolution that is the most difficult to explain. You may be able to convince him that natural selection can explain the appearance of complicated robots, who walk the Earth and write books and build computers, but you will have a harder time convincing him that a mechanical process such as natural selection could cause those robots to become conscious. Human consciousness is in fact the biggest problem of all for Darwinism, but it is hard to say anything “scientific” about consciousness, since we don’t really know what it is, so it is also perhaps the least discussed.

Nevertheless, one way to appreciate the problem it poses for Darwinism or any other mechanical theory of evolution is to ask the question: is it possible that computers will someday experience consciousness? If you believe that a mechanical process such as natural selection could have produced consciousness once, it seems you can’t say it could never happen again, and it might happen faster now, with intelligent designers helping this time. In fact, most Darwinists probably do believe it could and will happen—not because they have a higher opinion of computers than I do: everyone knows that in their most impressive displays of “intelligence,” computers are just doing exactly what they are told to do, nothing more or less. They believe it will happen because they have a lower opinion of humans: they simply dumb down the definition of consciousness, and say that if a computer can pass a “Turing test,” and fool a human at the keyboard in the next room into thinking he is chatting with another human, then the computer has to be considered to be intelligent, or conscious. With the right software, my laptop may already be able to pass a Turing test, and convince me that I am Instant Messaging another human. If I type in “My cat died last week” and the computer responds “I am saddened by the death of your cat,” I’m pretty gullible, that might convince me that I’m talking to another human. But if I look at the software, I might find something like this:

if (verb == ‘died’)
fprintf(1,’I am saddened by the death of your %s’,noun)
end

I’m pretty sure there is more to human consciousness than this, and even if my laptop answers all my questions intelligently, I will still doubt there is “someone” inside my Intel processor who experiences the same consciousness that I do, and who is really saddened by the death of my cat, though I admit I can’t prove that there isn’t.

I really don’t know how to argue with people who believe computers could be conscious. About all I can say is: what about typewriters? Typewriters also do exactly what they are told to do, and have produced some magnificent works of literature. Do you believe that typewriters can also be conscious?

And if you don’t believe that intelligent engineers could ever cause machines to attain consciousness, how can you believe that random mutations could accomplish this?

Comments
KF, Yes, I agree. The courtroom is a perfect example, where more information leads to certain conclusions; which may be left to the benefit of a doubt in certain circumstances. However, the basis for law in the first place, and all the arguments involved, appeal to the law of non-contradiction. Your color example shows that the process may not be exact, but conclusions can be made based on the revealed facts that one thing and not another occurred, or that one person and not another was responsible, based on sufficient evidence. Science seems to operate on the same principle, although with different data sets; so does philosophy, and so does mathematics.CannuckianYankee
September 22, 2010
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Vivid & GP: Thanks for the kind words. G PS: CY, the true logical opposite of white is not black, but not-white. Any shade of green, red, yellow or cyan will do. "Black and white thinking" is a strawman caricature argument, as you rightly point out.kairosfocus
September 22, 2010
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BarryR "There is no certainty expected anywhere in the process." Of course there's no certainty. But the court is in session because something happened, not nothing. I.e., the court's intent is to get as close to the facts of the something that happened. In other words, there is an indisputable fact behind the court's decision to convene, and that fact is that something as opposed to nothing happened. The court is there to determine whether someone is responsible for the something that happened. Behind the something is a reality that is a certain way. That we can't be certain the way that reality is does not determine that it is not a reality - it simply indicates that we don't know, and that's what we're there to find out beyond a reasonable doubt. We can be more certain with more facts presented, but we can't know for certain. I'm getting the impression that you think the way we perceive reality is the only way reality is - that our perceptions determine the way it is. And with that, it would be difficult to make a decision as to what reality is, because we are free to change our perception on issues in order to change the reality. With regard to the law of non-contradiction - it's not an issue of whether we see things as black or white. It's an issue of having a basis for anything that we can call truth. If we have no basis for truth, then we have no truth. That's not black or white thinking, but logical thinking. To believe that it's anything other than that is a contradiction.CannuckianYankee
September 21, 2010
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KF: I surely join vivid in my appreciation.gpuccio
September 21, 2010
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Kf Just want to thank you for all your contributions you make to this forum. Vividvividbleau
September 21, 2010
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GP: The Euler equation is hands down the most beautiful equation in all mathematics. And, it is so utterly astonishing, a surprise that comes out of nowhere, almost. Zero, the empty space on the abacus 1 the first number and the number of identity e, which governs so many kinds of growth and decay i, the strangest number of all: the "imaginary" root of a negative number. pi, the ratio of circumference to diameter of the "perfect" geometrical figure. All of them, tied up in one equation. If you needed a signature of the Ultimate Mathematician who built a world that embeds mathematics and number, here it is. Gkairosfocus
September 20, 2010
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kairosfocus @412, you have, once again, provided invaluable insights about the relationship between our rational minds and the rational universe that our minds apprehend.StephenB
September 20, 2010
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Thus we have have material 'particles' being 'destroyed' by teleportation of their transcendent information,,, A more convincing proof that transcendent information is foundational to material reality would be hard to imagine! Further evidence is abundant that the transcendent universal constants do not arise from any purported material basis and may in fact be called transcendent 'information' constants,,, As well Euler's number finds striking correlation to reality,,, The following verse and video are very interesting since, with the discovery of the Cosmic Microwave Background Radiation, the universe is found to actually be a circular sphere: Proverbs 8:26-27 While as yet He had not made the earth or the fields, or the primeval dust of the world. When He prepared the heavens, I was there, when He drew a circle on the face of the deep, Euler's Number - God Created Mathematics - video http://www.metacafe.com/watch/4003905 This following website has the complete working out of the math of Pi and e in the Bible, in the Hebrew and Greek languages respectively, for Genesis 1:1 and John 1:1: http://www.biblemaths.com/pag03_pie/ Michael Denton - Mathematical Truths Are Transcendent And Beautiful - Square root of -1 is built into the fabric of reality - video http://www.metacafe.com/watch/4003918 I find it extremely strange that the enigmatic Euler's number would find such striking correlation to reality. In pi we have correlation to the 'sphere of the universe' as revealed by the Cosmic Background radiation, as well pi correlates to the finely-tuned 'geometric flatness' within the 'sphere of the universe' that has now been found. In e we have the fundamental constant that is used for ascertaining exponential growth in math that strongly correlates to the fact that space time is 'expanding/growing equally' in all places of the universe. In the square root of -1 we have what is termed a 'imaginary number', which was proposed to help solve equations like x2+ 1 = 0 back in the 17th century, yet now, as Michael Denton pointed out in the preceding video, it is found that the square root of -1 is required to explain the behavior of quantum mechanics in this universe. The correlation of Euler's number, to the foundational characteristics of how this universe is constructed and operates, points overwhelmingly to a transcendent Intelligence, with a capital I, which created this universe! It should also be noted that these universal constants, pi,e, and square root -1, were at first thought to be completely transcendent of any material basis, to find that these transcendent constants of Euler's number in fact 'govern' material reality, in such a fundamental way, should be enough to send shivers down any mathematicians spine. This following video is very interesting for revealing how difficult it was for mathematicians to actually 'prove' that mathematics was true: Georg Cantor - The Mathematics Of Infinity - video http://www.metacafe.com/watch/4572335 entire video: BBC-Dangerous Knowledge (Part 1-10) http://www.youtube.com/watch?v=Cw-zNRNcF90 As you can see, somewhat from the preceding video, mathematics cannot be held to be 'true' unless an assumption for a highest transcendent infinity is held to be true. A highest infinity which Cantor, and even Godel, held to be God. Thus this following formal proof, which was referred to at the end of the preceding video, shows that math cannot be held to be consistently true unless the highest infinity of God is held to be consistently true as a starting assumption: Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century Excerpt: Gödel’s Incompleteness Theorem says: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle - something you have to assume to be true but cannot prove "mathematically" to be true.” http://www.cosmicfingerprints.com/blog/incompleteness/ THE GOD OF THE MATHEMATICIANS - DAVID P. GOLDMAN - August 2010 Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel's critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes. http://www.faqs.org/periodicals/201008/2080027241.html This following site is a easy to use, and understand, interactive website that takes the user through what is termed 'Presuppositional apologetics'. The website clearly shows that our use of the laws of logic, mathematics, science and morality cannot be accounted for unless we believe in a God who guarantees our perceptions and reasoning are trustworthy in the first place. Proof That God Exists - easy to use interactive website http://www.proofthatgodexists.org/index.phpbornagain77
September 20, 2010
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Of note: An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. Ions have been teleported successfully for the first time by two independent research groups Excerpt: In fact, copying isn't quite the right word for it. In order to reproduce the quantum state of one atom in a second atom, the original has to be destroyed. This is unavoidable - it is enforced by the laws of quantum mechanics, which stipulate that you can't 'clone' a quantum state. In principle, however, the 'copy' can be indistinguishable from the original (that was destroyed),,, http://www.rsc.org/chemistryworld/Issues/2004/October/beammeup.asp Atom takes a quantum leap - 2009 Excerpt: Ytterbium ions have been 'teleported' over a distance of a metre.,,, "What you're moving is information, not the actual atoms," says Chris Monroe, from the Joint Quantum Institute at the University of Maryland in College Park and an author of the paper. But as two particles of the same type differ only in their quantum states, the transfer of quantum information is equivalent to moving the first particle to the location of the second. http://www.freerepublic.com/focus/news/2171769/postsbornagain77
September 20, 2010
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Seeing that Euler's Identity was referenced as well as the fact someone made the absurd claim that material is primary to determining reality in math. I would like to answer as strongly as possible that there is no 'material' basis to reality. All reality in fact reduces to information or what I prefer to call "The Word" of John 1:1: notes: It is very interesting to note in this following paper that the quantum state of a photon is actually defined as 'infinite information': Explaining Information Transfer in Quantum Teleportation: Armond Duwell †‡ University of Pittsburgh Excerpt: In contrast to a classical bit, the description of a (photon) qubit requires an infinite amount of information. The amount of information is infinite because two real numbers are required in the expansion of the state vector of a two state quantum system (Jozsa 1997, 1) --- Concept 2. is used by Bennett, et al. Recall that they infer that since an infinite amount of information is required to specify a (photon) qubit, an infinite amount of information must be transferred to teleport. http://www.cas.umt.edu/phil/faculty/duwell/DuwellPSA2K.pdf These following studies verified the violation of the first law of thermodynamics that I had suspected in quantum teleportation: How Teleportation Will Work - Excerpt: In 1993, the idea of teleportation moved out of the realm of science fiction and into the world of theoretical possibility. It was then that physicist Charles Bennett and a team of researchers at IBM confirmed that quantum teleportation was possible, but only if the original object being teleported was destroyed. --- As predicted, the original photon no longer existed once the replica was made. http://science.howstuffworks.com/teleportation1.htm Quantum Teleportation - IBM Research Page Excerpt: "it would destroy the original (photon) in the process,," http://www.research.ibm.com/quantuminfo/teleportation/ Unconditional Quantum Teleportation - abstract Excerpt: This is the first realization of unconditional quantum teleportation where every state entering the device is actually teleported,, http://www.sciencemag.org/cgi/content/abstract/282/5389/706 Of note: conclusive evidence for the violation of the First Law of Thermodynamics is firmly found in the preceding experiment when coupled with the complete displacement of the infinite transcendent information of "Photon c": http://docs.google.com/Doc?docid=0AYmaSrBPNEmGZGM4ejY3d3pfMzBmcjR0eG1neg More supporting evidence for the transcendent nature of information, and how it interacts with energy, is found in these following studies: Single photons to soak up data: Excerpt: the orbital angular momentum of a photon can take on an infinite number of values. Since a photon can also exist in a superposition of these states, it could – in principle – be encoded with an infinite amount of information. http://physicsworld.com/cws/article/news/7201 Ultra-Dense Optical Storage - on One Photon Excerpt: Researchers at the University of Rochester have made an optics breakthrough that allows them to encode an entire image's worth of data into a photon, slow the image down for storage, and then retrieve the image intact. http://www.physorg.com/news88439430.html This following experiment clearly shows information is not an 'emergent property' of any solid material basis as is dogmatically asserted by some materialists: Converting Quantum Bits: Physicists Transfer Information Between Matter and Light Excerpt: A team of physicists at the Georgia Institute of Technology has taken a significant step toward the development of quantum communications systems by successfully transferring quantum information from two different groups of atoms onto a single photon. http://gtresearchnews.gatech.edu/newsrelease/quantumtrans.htm The following articles show that even atoms (Ions) are subject to teleportation:bornagain77
September 20, 2010
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---Markf: "I think everyone has aired their views and no one has changed their mind about anything. So I will leave it there." Does this mean that you are not goint to tell why, in your words, it is "nonsensical" to say that Jupiter can both exist and not exist at the same time?StephenB
September 20, 2010
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#411 Gpuccio Sorry about your personal problems. I am a bit tired of this particular debate. I think everyone has aired their views and no one has changed their mind about anything. So I will leave it there. Markmarkf
September 20, 2010
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KF: that equation is really amazing. I believe most mathematicians are neo-platonists regarding their discipline. I became aware of the neo-platonists model of mathematics through the books of Penrose, who has certainly influences me very much, especially with his argument about Godel's theorem and the nature of human cognition. The beauty of mathematical constructions is in itself surprising, but their usefulness in understanding physical reality is really beyond any "natural" explanation: it definitely points to a parallel structure in physical reality and in our personal minds.gpuccio
September 20, 2010
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EZ: The complex frequency domain [without and with damping as an issue] is where signal processing lives. And as to the reality or not of numbers, I note just one little equation: 0 = 1 + e ^(i*pi) -- Euler Believe it or not, the five most important numbers in mathematics are tied together, through the complex domain. But more to the point, numbers are a real aspect of experienced reality, but have no physical being as such. You cannot show 0, 1, or 2, etc, save as a property that we symbolise using numerals. But that mysterious, shadowy set of properties and their relationships, turn out to be central to reality, as physics and related disciplines show. And that points, ever so subtly but strongly, to a world of reality beyond the immediately physical. You may resist it, but there the compass needle points. (And every time you tune to an FM station, you are looking at the reality of how intelligible signals are constructed form combinations of waves at different frequencies, as the Fourier transform and integral point to. Every time you see something undergo damped harmonic vibrations, it is pointing in the same direction, with a damping factor added. Indeed, the relevant transfer functions strongly depend on poles in the complex frequency domain of INFINITE amplitude. That's part of the rubber sheet model of frequency response, and it is the technique I taught my students to visualise the otherwise utterly abstract reality. I will not get into Z-transforms here, save to say that with digital filters, they are unit delay elements, and can essentially replicate anything that can be done in the analogue world, and more. That's before we look at how such models apply to say economics.) GEM of TKIkairosfocus
September 20, 2010
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Mark: I must thank you for your answers at 262 and 263. And apologize that only now I have been able to read them (some personal problem, now solved :) ). I still would like to stay out of all the discussion, but I will try to give you some simple description of what I think. I think that we, as humans, have cognitive powers which are based on some universally shared faculty, however we can call it. That faculty is mainly intuitive, and guides us in making and comparing statements. That intuitional cognition is therefore largely innate, and not experience derived, and is the basis for all mathematical and logical thinking, and indeed for all thinking. So, I am in the group of neoplatonists for what regards mathematics. The really astonishing fact is that such intuitive and innate faculty, and its products (logic, mathematics, empirical inference, and so on) are so powerful in understanding reality. Empirical sciences are continuous evidence that our mental models are definitely useful in explaining the reality out of us. Indeed, there is no a priori reason for that. It's definitely an amazing mystery. And a mystery which works even at very counter-intuitive levels: QM is good evidence that very abstract elaborations of our minds and of pure mathematics can guide us to conclusion that we would have never reached by other ways, and which work: outer reality seems strangely to correspond to our mathematical constructs about it. That said, I can agree with you that these problems are very subtle, and that we can define and build our mental products in very different ways. But, if there were no shared intuitions in our minds, that would only amount to a lot of changing local conventions, devoid of any universality and usefulness.gpuccio
September 20, 2010
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BarryR,
Really? That would mean I was certain that I should be using a continuum. And I’m not certain about that — I have no problem admitting this approach may be wrong. It just happens to be the best one I’ve come up with so far.
In a few words, you seem certain that you should remain either uncertain or certain, depending on whatever you deem worthy in your criteria. You seem certain that there is such a thing as certainty and uncertainty. You seem certain that there really is such a thing as criteria. You seem certain that there is such a thing as you. You seem certain that the phrase "I am not certain" actually means what you want it to mean, and that I shouldn't change it to mean blue rabbits love chocolate. You seem certain that every word of the phrase "I am not certain" has individual and particular meaning, and can be used together to form a coherent sentence. You seem certain that you're typing these words on a screen. I could go on and on. I am surprised I have this much patience, normally I don't feed condescending trolls, especially ones that are self-triumphant (a tactic that reminds me of the minister of Iraq claiming that there was no American invasion when we were pulling up outside Baghdad while he was talking to reporters) when there is no victory. You seem to be certain that something like a continuum is "better" than a binary approach to truth. You seem certain that this is not tantamount to saying that there is no black and white truth, that it is all a shade of gray, (as if this statement were not asserted as a black or white statement itself). The endgame is that if you're near certain about anything, it is only by a fixed proximity of real certainty that you can get "near" to it. If the train station were just as mobile as the train, you could never get nearer to your destination. If certainty were not itself an actuality, you couldn't get nearer to it.Clive Hayden
September 20, 2010
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So glad I didn't have to resort to physical models like that!! :-) I don't see how complex numbers point beyond materialism since they don't 'exist' (even though they can be measured and manipulated). Mathematics is not philosophy thank goodness. And if your mathematical models don't accurately reflect the real, material world then the model is wrong and needs to be revised. So, for me, at least as far as applied mathematics is concerned (and control theory is a particularly useful area) the mathematics is a slave to the material. And the fact that a cubic equation has three solutions, in most real-world situations it's only the real ones that count. It's worth looking at the ones that don't seem to have an observable parallel but topology was just an abstract field until relativity suggested a way it could be used.ellazimm
September 20, 2010
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EZ:
Stuff that isn’t real [PHYSICALLY MANIFEST] can still be useful and have meaning in context
That is, I am pointing out the gap between reality and physical materiality. How ironic it is that right at the heart of science is number, and number, as NWE rightly corrected Wiki in an ever so eloquent and rich definition, is:
A number is an abstract mathematical object represented by a symbol that is used in counting and measuring. A symbol that represents a number is called a numeral.
So, right in the heart of the central citadel that the evolutionary materialists think they hold lies something that ever so plainly points beyond materialism. And, BTW, I used to teach my controls students to spot the complex domain poles for dynamical objects, based on their time domain transient responses, using Laplace transforms (based on the heavy rubber sheet with propped up poles and nailed down zeros model; love my 3 edns of Ogata). Dose "imaginary" objects does be very, very real! (And not just for flow fields and systems governed by dynamical ODEs. They also relate to something as mundane as the bandwidth of an FM signal, per Fourier.) GEM of TKIkairosfocus
September 20, 2010
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KF: I wasn't trying to prove imaginary numbers exist! I was just saying that the construct is useful for modelling real life!! It's a a mental construct obviously. And i (SQRT(-1)) IS the solution to an algebraic equation: x^2 +1 = 0. When you allow that construct then all polynomial equations with real coefficients have solutions but in the complex numbers. And the number of solutions is equal to the degree of the equation with the exception of the trivial cases. Material objects don't have cardinality of zero, clearly. I wasn't suggesting they do! I'm not sure why you're arguing with me. The set of all subsets of a given set includes the empty set but that's not an object! I could say that a collection of objects containing no objects has a cardinality of zero but that's not what you're asking for. If you take a line segment, removed the middle third, take out the middle third of the remaining pieces and continue that to infinity you will get a non-empty collection of points which has a measure (length) of zero but again that is no a real object. You can do similar things with two and three dimensional 'objects' like Sierpinski gaskets and sponges. I concede! I was only pointing out that it is possible to construction mental . . . . NEVERMIND!! I'm not disputing that two object plus two more objects gives you four objects!! But if you want to analyse the flow of a fluid across a surface you might find it useful to use a complex valued function which, when integrated, gives you two terms (one of which is the coefficient of the imaginary part) which indicate two aspects of the fluid flow. Stuff that isn't real can still be useful and have meaning in context!!ellazimm
September 20, 2010
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Zero: the number of apples after you have eaten all of them?DiEb
September 20, 2010
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EZ: Kindly show me a material object that exhibits cardinality of zero. (A pair of empty braces, one of he symbols for the empty set, will not do: {} ) GEM of TKI PS: Similarly, show me the number SQRT (-1) as a material entity. (I did not say show me its manifestations in action, I am asking you to show me a material --physical -- object that exhibits the number in question. An Argand diagram, a 2-dimensional graph, will not do.)kairosfocus
September 20, 2010
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F/N: Just to remind us from that hostile witness making an admission against interest, Wikipedia: ________________________ >> A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. Certain procedures which take one or more numbers as input and produce a number as output are called numerical operations. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. More common are binary operations which take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic. >> _________________________ Notice that ever so delicately circular term "mathematical object"? Here is NWE's take:
A number is an abstract mathematical object represented by a symbol that is used in counting and measuring. A symbol that represents a number is called a numeral.
Number itself cannot be seen, touched, felt, tasted, smelled, held, pinched, or heard. It is not tangible or concrete. It is indeed abstract, a conceptual entity we infer from experience. It is a mental construct, whihc we manipulate using symbols and symbolised operations. Thence, it is reflective of the conscious, thinking, knowing reasoning mind. But, Wiki, ever so reliably materialistic wiki, cannot acknowledge that. So, that little part is conveniently left out. And that is why EZ above felt it so necessary to highlight how we find that collections of material objects manifest numerical properties. Guess what, so do souls: we are unified self-aware selves, i.e. we are ONES. GEM of TKIkairosfocus
September 20, 2010
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EZ: 1: None of those things changes the fact that when we have collections, we see the following property of their cardinality: 2 + 2 = 4 2: Similarly, that cardinality has real properties such that we see that not only does 2 + 2 = 4 in fact, but it must do so. 3 : Going beyond that, from the operation +, we derive all other relevant arithmetic operations, and once we respect its meaning, we will see that 2 + 2 = 4 is a self-evident truth. 4: You are operating in an environment where various distractive devices have been used to try to persuade us that we can overlook, ignore or deny the basic, self-evident truth: 2 + 2 = 4. GEM of TKI PS: Kindly show me how number is a material entity, as opposed to how a collection of material objects will have a property, their number. (In short two guavas plus two guavas, as I can see by walking out into my backyard, will give me four guavas. But the property that a two set joined to another two-set yields a four-set is not itself a material property.)kairosfocus
September 20, 2010
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---BarryR: "So while I see no difficulty creating a consistent formal system where Jupiter simultaneously exists and does not exist, my (tentative) conclusion is that such a system is useless." But you did not answer my question. Is it possible for Jupiter to exist and not exist at the same time? There are only three possible answers. Yes, no, or I don't know because I don't have (or acknowledge) the necessary tool of logic [the law of non-contradiction] that would justify a definitive answer. Since you will not assert a yes or a no, I have to assume that the answer is--"I don't know"--meaning that your system of thought cannot generate an answer. What good, then, is your thought system if it cannot affirm that which everyone else knows to be the case: Jupiter cannot exist and not exist at the same time. Further, inasmuch as you propose that the "principle" of non-contradiction is not a universal law and therefore is not "useful" in all contexts, and given that facts in evidence can provide no answer to the question, how do you know where to apply or or when it would be useful? Put another way, what is your standard for deciding when and where it fits?StephenB
September 20, 2010
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I wasn't arguing with the notion that two material things plus two more material things gives you four things!! I was just saying that it is possible to define (and use) systems where some mathematical results look counter-intuitive. No biggie. And I didn't even get to surreal numbers, hyper-real numbers or different sizes of infinity!! Or fractional dimensions.ellazimm
September 20, 2010
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PPPS: How could I forget pounds, shillings pence guineas; inches, feet, yards, chains, furlongs, miles, leagues, grains, drams, ounces, pounds [mass, not force!] troy and avoirdupois, slugs, cups, pints, quarts, gallons [and recall, it's not just Wine/US vs Imperial], square feet, square chains ["tasks" of land], acres, square miles etc, and all that wonderful menagerie of measurements and units that require ever so many odd steps to move up and down? Try CGS vs SI in science, too. None of these changes the basic self evident fact that: || + || --> ||||kairosfocus
September 20, 2010
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PPS: EZ [and RB], it might help to know that I cut my eye-teeth on a fondly remembered HP 21, and changed milk teeth on the 6800/6809 MPU family, programmed at assembly/machine code level. RPN is my preferred calculator operating system, and a copy of XCalc set to ENG notation sits on my desktop as convenience calculator. Modulo-arithmetic, boolean algebra, clock arithmetic, complex numbers, complex exponents, and vectors, phasors, the right ascension and declination, etc are all familiar old friends. None of them affects the basic self evident fact that || + || --> ||||kairosfocus
September 20, 2010
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PS: Clock arithmetic [cf. angle arithmetic on the degree system] is rooted in the sexagessimal system, with a duodecimal one added on for larger units, probably because of how many factors 12 has. I think astronomers did calculations in base 60 until just about the days of Newton; think of how many factors evenly fit into 60. [When I used to mention the base-60 system in a first lecture on digital electronics, my students used to get a kick out of the name.]kairosfocus
September 20, 2010
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EZ: 1: Irrelevant -- that you can create systems in which you redefine + to be modulo etc does not change:
|| + || --> |||| 2 + 2 = 4
2: Distractive -- you are dragging that old red herring away to a strawman. In slightly more details: a --> What has been discussed is the self-evident truth that 2 + 2 = 4. b --> In this context, it is understood on our experience of the world [use match sticks], and the symbols 2, +, = and 4 take their common garden variety meanings we learned at age 5 from Ms Smith with the knuckle-smacking ruler. c --> The operation of addition is putting together, and once we do that two 2-sets cojoined yield a 4-set. d --> And, that MUST be so, given the way we recognise cardinality; in effect counting/matching to the standard ordered set to exhaustion, on or the equivalent. e --> Also, we are not defining anything here, we are observing an operation and its result, summarising the consequence for cardinalities. [Cf 187 - 191 above.] f --> Thus, 2 + 2 = 4 is an example of a self-evident truth. As is "Error exists." And as are many more cases. _________________ I suggest you should think again, very carefully, on why you find such a "need" to manufacture an objection to such an elementary and easily demonstrated truth. [Cf. 362 above on where that problem points.] GEM of TKIkairosfocus
September 20, 2010
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Angle measure is in mod 360 degrees or 2pi radians or 400 gradians. So that: 285 + 200 = 125 degrees. Not taking into account the rotations just the ending position. Also, rotational symmetries in shapes can be described in modular arithmetic. I'm not even going to get into imaginary numbers (that include the square root of -1) which end up having applications in the real world. That is just really weird. But it works!!ellazimm
September 20, 2010
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