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
[Pi is a universal constant in Eucledian space.] ---markf: "That is true – but it only holds where the axioms of Eucledian space hold." It is a law for which there are no exceptions. ---In non-Eucledian spaces the ratio of the circumference to diameter of a circle can almost anything." Non-Euclidlian geometry is a different subject matter, which is one more reason why the word absolute applies and the word universal is problematic. To the extent that the laws of non-Euclidian geometry have been discovered, they are as absolute and unchangeable as the laws of Euclidian Geometry. Why would anyone expect the laws of plane geometry to be similar to the laws of spherical geometry? That, by the way, is why BarryR injected the word "universal" in the discussion, to shift the discussion away from the absolute, unchanging laws in the various mathematical discipines and onto the trivial and obvious point that no universal law can speak to all disciplines at the same time. Hence, my insistence on the word absolute. ---"The real question is whether there is anything special or absolute about the statements of Eucledian geometry as opposed to any other." Obviously, mathematical laws are absolute. Do you know of any that have been found to be untrue? They certainly work in the real world where they are applied every day. Do you know of any instances in which a bridge collapsed because the laws of trigonometry failed to hold up?StephenB
September 13, 2010
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Cabal@181 Pi is both irrational (cannot be expressed as a ratio of two integers) and transcendental (not a root of a non-constant polynomial equation with rational coefficients). Different systems handle this differently. Constructivists hold that only accepted proof of a numerical object is the construction of that object, and so while approximations to irrationals are known to exist, there is no proof for irrationals. This ends up giving you an arbitrarily tight bound on pi, but you can't get to pi itself. And, of course, pi won't show up in any number systems that can't handle irrationals (like the mathematics that describe how your computer operates --- that uses a very loose approximation). Also, any system that is sufficiently simple to escape Godel's incompleteness theorem probably isn't complex enough to construct irrationals, but that's just an intuition.BarryR
September 13, 2010
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---markf: "The prescence of rules and deductions called “laws” is surely not the issue. I accept that there are laws of chess. I just don’t think they apply outside of chess." You are making the same mistake that BarryR makes. There are no laws of chess, only rules. Laws are discovered; rules are established. As I have already indicated with my brief reference to the law of sines/cosines, the laws of mathematics were discovered and are unchangeable. The word "absolute" covers that better than the word "universal," though the latter term would also apply. Few people, however, would think to use that word, which is why I objected to BarryR's attempt to inject it into the discussion for purposes of measuring its frequency of use over the internet--as if that means anything. He, like you, are not just making an error in interpretaion. You are both wrong about the facts in evidence.StephenB
September 13, 2010
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#177 I guess the odd symbol was pi? The key point here comes from vividbleau's comment: Pi is a universal constant in Eucledian space. That is true - but it only holds where the axioms of Eucledian space hold. In non-Eucledian spaces the ratio of the circumference to diameter of a circle can almost anything. The real question is whether there is anything special or absolute about the statements of Eucledian geometry as opposed to any other.markf
September 13, 2010
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Onlookers: You will observe how RB has again, sadly, evaded the issue. We deal with a real world, where we can collect objects and group them — go get a match full box for illustration.
Being an onlooker, I suppose i am entitled to make a comment here. It seems to me that much of kf's argument is about counting discrete objects. Nobody in his right mind would ever claim that two apples plus two apples mysteriously might reappear as onehundredandfortyseven (147) apples. That's a feat of biblical proportions. The question of ¶ is another matter; we are dealing with a particular relationship that cannot be expressed with a whole number. I'd have to study the subject first but I suspect maybe nobody knows if an exact number may be found this side of an infinite number of decimals. Hope the last sentence makes sense.Cabal
September 13, 2010
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#175 Inasmuch as I said nothing about “universal” mathematical laws, I can’t imagine what you mean? I said that there are mathematical laws, and you said that there are none. That is the issue on the table If a mathematical law is not universal then presumably it does not apply at all times and all places (or do you mean something else by "universal"? If all you are claiming is that there are some mathematical laws that are true under some conditions, I don't think Barry or I would disagree. We already discussed that 2+2=4 when applied to apples. The prescence of rules and deductions called "laws" is surely not the issue. I accept that there are laws of chess. I just don't think they apply outside of chess.markf
September 12, 2010
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StephanB@176
Inasmuch as I said nothing about “universal” mathematical laws
That's my formulation, yes. I thought you disagreed. Is that correct?
I said that there are mathematical laws, and you said that there are none. That is the issue on the table.
I'm pretty sure you still don't understand my actual position, but I don't think there's anything to be gained by repeating it.
You claim that your authors agree with you, but when I ask you to show me when and where, you opt out.
I think I showed you that, for these laws being universal and all, nobody seems to consider them worth writing about. But since you insist: Bruce Edmonds, What if /all/ truth is context-dependent, draft, 2001.
Pure mathematics aspires to a world of its own. It is concerned with what can be formally proven given certain structures, assumptions, etc. For example, given Peano's axioms for arithmetic, the standard notation and some standard logical inference operations, one can prove the statement "1+1=2". Does this not mean that "1+1=2" is a universal truth, devoid of context? I would argue not.
I don't hold out any great hope that you're going to read the paper. Nor do I hold out any great hope that you're going to be able to supply any citations for your side, much less supply any universal / objective mathematical laws.
On the other hand, I can certainly cite a number of authors and mathematicians who agree that there are, indeed, a number of mathematical laws.
Before you waste you time, I certainly believe there are mathematical laws; they're just context-dependent. I'm pretty sure you don't understand the distinction, and frankly I don't know how to explain it to you.
general LAW of sines
How does the law of sines apply to graph theory? I don't think it does. If you're working with a geometric system, sure, then within that system you can describe a general law. But only within that system. I'm sorry you've put so much time into this and still don't understand what's at issue.
Do you dispute any of this?
I don't dispute that you've found yet another context-dependent law. They're rather thick on the ground, you know. I will dispute that these are context-independent. Tell you what: After you've read Edmonds let me know and we can discuss his formulation, and I'll refrain from any other replies to you until you do so.BarryR
September 12, 2010
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kairosfocus@174 You've demonstrated how addition holds when mapped to discrete objects that remain discrete under addition. As that doesn't even describe all physical systems, I'm not clear why you think this is universal.BarryR
September 12, 2010
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vividbleau@172 Well done. That's the most mathematically sophisticated reply I've gotten in this thread. (No, that's not sarcasm.) But as you say:
Pi is a universal constant in Eucledian space.
I'd go so far as to say pi will be found in any system that supports transcendental numbers. But as not all systems support transcendental numbers, I wouldn't consider pi to be a universal constant. Still, as candidates go, it was much better than "exponentiation".BarryR
September 12, 2010
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---BarryR: "I’d like to think you didn’t make up the idea of universal mathematical laws. Can you give me the citation where you first heard of it?" Inasmuch as I said nothing about "universal" mathematical laws, I can't imagine what you mean? I said that there are mathematical laws, and you said that there are none. That is the issue on the table. You claim that your authors agree with you, but when I ask you to show me when and where, you opt out. Why cite them if you cannot show how they support your claim? On the other hand, I can certainly cite a number of authors and mathematicians who agree that there are, indeed, a number of mathematical laws. Just to make sure that I understand you, are you saying, for example, that the law of sines/cosines is not a law in spite of the fact that its discoverers called it a law and that all mathematicians that I know of would call it a law? According to Wikipedia, "The spherical LAW of sines was discovered in the 10th century. It is variously attributed to al-Khujandi, Abul Wafa Bozjani, Nasir al-Din al-Tusi and Abu Nasr Mansur Al-Jayyani's The book of unknown arcs of a sphere in the 11th century introduced the general LAW of sines." "The plane LAW of sines was later described in the 13th century by Nas?r al-D?n al-T?s?. In his On the Sector Figure, he stated the LAW of sines for plane and spherical triangles, and provided proofs for this LAW." Do you dispute any of this? Frankly, I don't understand how you can continue on with this easily refuted proposition of yours.StephenB
September 12, 2010
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PS: Re Vivid, Pi is the ratio of the circumference of a circle to its diameter. That this is a fixed number has not changed, nor will it change, not so long as circles remain circles, similar to the traces we make by whirling a compass. This is a brute fact of reality, and it is certainly an unchanging mathematical truth. We may not be able to specify exactly pi in any system of fractions, but that does not change the reality that a circle has certain key properties.kairosfocus
September 12, 2010
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Onlookers: You will observe how RB has again, sadly, evaded the issue. We deal with a real world, where we can collect objects and group them -- go get a match full box for illustration. Cluster matches like so: { | | } { | | } Recluster, by pushing together: { |||| } You have just physically instantiated the arithmetic binary operation, + for two two-sets [symbolised by 2], and yielding a four-set, symbol 4. 2 + 2 = 4 Based on the meaning of 2, +, = and 4 as conventional symbols [which can vary, just make sure you are using well thought out symbols], this is not just inductively true, nor is it simply an arbitrary game of symbols in a system that rests on ultimately circular tautologies. No, instead we are here again up against the reality of self-evident truth. That is, given our experience of the world as conscious and reasoning creatures, providing we have a good understanding of 2, +, = and 4 [all of which inextricably interact in their meaning], it does not just happen to be so that 2 + 2 = 4, but that it MUST be so. On pain -- pardon, but this is part of what a self-evident truth is -- of immediate descent into obvious absurdities. In the case where someone attempted to redefine that 2 + 2 = 147, there are a great many absurdities involved. Starting with arbitrary redefinition of symbols used in an evasive equivocation. And, going on to the situation where we are now seeing + being sued in at least two different ways without means to distinguish, i.e. communication and reason are beginning to break down. Hardly less significant, 147 is a notation: 1 x 100 + 4 x 10 + 7 x 1. See how the original meaning of 4, which includes properties like 4 = 2 + 2, or 1 + 3, or 1 + 1 + 1 + 1, has resurfaced? [If, in attempting to deny or dismiss a point you find you are implicitly assuming and using it, then that is a strong sign that you are in error.] Axiomatic systems in mathematics may be constructed, but the facts that they address are real, and in the case of the sort of fairly simple cases like basic arithmetic, are often self-evident. GEM of TKIkairosfocus
September 12, 2010
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typo I knowvividbleau
September 12, 2010
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RE 168 "But nothing listing what these universal mathematical laws might be." Pi is a universal constant in Eucledian space. Vividvividbleau
September 12, 2010
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CH@170
Of course your cats and my dog have consciousness, and know things about the universe, isn’t that obvious?
I sure thought it was. Granville Sewell (who authored the article to which we're responding) doesn't find it obvious at all. For him, human consciousness is special, not just further along on a continuum of consciousness. I'm glad we've found something to agree on.
You can choose axioms depending on what you’re trying to accomplish, as you can choose material for building a house, neither one contradicts or suspends the laws that govern them, such as basic arithmetic and physics.
We may finally be converging. I'm free to build a house out of diet coke cans --- it won't be useful, but it can certainly be done. I'm also free to choose axioms such that 2+2=147 --- also not very useful, but neither is it invalid. Are we in agreement?
I was referring to your false dilemma of “different logics”, as if they were conventional, like choosing to drive on the right side of the road instead of the left.
Yes, which logic you use (and there are several, several of which are mutually exclusive) is your choice. This follows from the fact that logic, as in any branch of mathematics, is derived from axioms, and we can chose our axioms to suit the problem we're trying to solve. In fewer words, just as there is no universal rule stating that 2+2=4, there is also no universal rule defining the logical operators "and", "or", and "not".
If I had to compare a computer programmer and an Oxford don who received three firsts in his education and who Cambridge invented a chair for him, I think I’d choose the latter, if you’re going to play the credentials game. Yours is nil compared to Lewis’s.
Interesting. I appear to have had more training in philosophy than C. S. Lewis. (Enough classes to qualify for a minor during my undergrad as well as several graduate classes in logic when I was getting my Masters.) I hadn't considered taking a few philosophy classes sufficient to be considered "trained as a philosopher". I certainly don't consider myself a philosopher and Lewis didn't consider himself a philosopher either. Putting that to one side, Lewis isn't wrong because he's not a philosopher, he's wrong because he's making several mistakes that actual philosophical training would have prevented. Part of my training was in learning how to spot those kinds of errors. That's part of the difference between reading Plato as Literature and reading Plato as Philosophy. (You'd probably know this: did he ever submit any of his philosophical work for peer review? I can't find anything, but I haven't looked too hard.)
We’re far adrift from your explanation as to why Chesterton was wrong in his critique of scientism and his explanation of the problem of induction, which you never addressed.
Tell you what: remove the moderation block and I will go through and give you a point by point critique of Chesterton. If I'm going to put in the time to do this, I want some assurance that the post will actually show up here.BarryR
September 12, 2010
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BarryR,
It does? Interesting. My cats know many things about the universe, but I didn’t think you be comfortable ascribing “thoughts” to them (as the capacity for thought would certainly imply consciousness, yes?) So I can go either way. Either my cats manage to know the universe without thoughts, and the above is correct; or my cats possess some level of consciousness and use thoughts to know the universe.
Of course your cats and my dog have consciousness, and know things about the universe, isn't that obvious?
I’m not finding anyone who agrees with you, and several people who disagree, so for the moment I’m going to provisionally assume that choice exists. For example: Of course, “the activity o mathematics is not just randomly writing down formal proofs for random theorems”, because “the choices of axioms, of problems, of research directions, are influenced by a variety of considerations — practical, artistic, mystical”…
That's interesting, because I was responding to your first order and second order logic argument, not whether axioms can be chosen with regard to mathematics for certain purposes (which doesn't, by the way, negate that 2+2=4). You can choose axioms depending on what you're trying to accomplish, as you can choose material for building a house, neither one contradicts or suspends the laws that govern them, such as basic arithmetic and physics. But this really just changes the subject, as I was referring to your false dilemma of "different logics", as if they were conventional, like choosing to drive on the right side of the road instead of the left. As if making sense of how big is yellow could really be, depending on my own invention of logic. And you're simply wrong about Lewis not being trained in philosophy: Following the end of the war in 1918, Lewis returned to Oxford, where he took up his studies again with great enthusiasm. In 1925, after graduating with first-class honors in Greek and Latin Literature, Philosophy and Ancient History, and English Literature. From January 1919 until June 1924, he resumed his studies at University College, Oxford, where he received a First in Honour Moderations (Greek and Latin Literature) in 1920, a First in Greats (Philosophy and Ancient History) in 1922, and a First in English in 1923. From October 1924 until May 1925, Lewis served as philosophy tutor at University College during E.F. Carritt's absence on study leave for the year in America. Update wikipedia if you'd like. There is no shortage of actual and true biographies of the man readily available online. So that should clear up your ad himonem against Lewis not being trained in philosophy. And secondly, you're not trained in philosophy, so why should I regard anything you have to say by the same criteria? If I had to compare a computer programmer and an Oxford don who received three firsts in his education and who Cambridge invented a chair for him, I think I'd choose the latter, if you're going to play the credentials game. Yours is nil compared to Lewis's.
When we observe this behavior in others, we usually classify it as instinct.
Instinct is innate, not knowledge about the world, but an impulse, like an appetite. An appetite is not knowledge about food. We're far adrift from your explanation as to why Chesterton was wrong in his critique of scientism and his explanation of the problem of induction, which you never addressed. Real laws of logic and reason and mathematics, necessary relations between things, and why we can see why they are necessary, not just that they exist together, are fundamentally unlike anything we ever observe is nature by virtue of seeing two things together. I'm still waiting for your answer and argument that proves this to not be the case.....Clive Hayden
September 12, 2010
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kariosfocus@167
But in all cases, one is very careful to distinguish contexts so one avoids errors of equivocation leading to gross contradiction.
You've hit on the issue precisely. The universal laws that several posters here wish existed are not susceptible to context. 2+2=4 is not a universal truth, it is a truth within a specific mathematical context formed by a specific choice of axioms.
(BTW, does anyone say “Topology” much anymore?)
I find topology fascinating but, once I get past the introductory examples, completely counterintuitive. This is odd, as there's a pretty good mapping to graph theory and that's pretty easy. So yes, that's the reason for my reluctance to use topological examples.BarryR
September 12, 2010
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StephanB@165
Apparently, you have been so steeped in postmodern subjectivism, that you cannot distinguish between socially constructed rules and objective laws.
It's unfortunate that 19th C. mathematicians who invented abstract algebra couldn't make that distinction either. It's also a distinction that geometers in the 19th C. were unable to make when they came up with non-Euclidean geometry. Carl Friedrich Gauss failed to make the distinction back in 1801 when he published the first work on modular arithmetic. I'm afraid the entire history of mathematics is nothing but a sorry parade of ignoring universal truth in favor of inventing cool stuff that works.
Irrelevant. Mathematicians are not free to pick and choose which mathematical laws they will honor and which ones they will not.
I've provided citations to the peer-reviewed literature showing exactly that in addition to the specific examples of modular arithmetic and non-standard analysis. Why should I believe you over professional mathematicians?
Do you have an example of an unchangeable law from graph theory? I’m not aware of any, but then you may know more about the topic than I do.
Irrelevant.
Your argument isn't helped by the fact that you're not able to come up with any universal laws of mathematics. Exponentiation certainly wasn't one and you don't know any from graph theory. I'd find it much easier to believe in a universal mathematical law if you were able to write down for me.
Perhaps you can provide a quote from one of your authors indicating that mathematics has no laws.
Ah, yes, *I* have to provide citations, but you don't. I think you'd agree that asking me for a citations showing there were no universal piglets of mathematics would be a little difficult to fulfill. You're asking me to provide evidence for a negative statement, and that's usually seen as poor debating technique. So while I cannot give you any evidence that no universal piglets of mathematics exist, I can give you evidence that universal piglets are something that mathematicians don't spend a lot of time thinking about. scholar.google.com returns the following hit counts: 881,000 "calculus" 7,490 "modular arithmetic" 4,820 "nonstandard analysis" 16 "universal laws of mathematics" 10 "universal mathematical laws" 0 "universal piglets of mathematics" Of these sixteen, we have a biography of Descartes, a paper on free will, a review of a Descartes biography, a short piece at answers.com, a book called "The philosophy of left and right"... But nothing listing what these universal mathematical laws might be. As far as mathematicians publishing mathematics goes, "universal mathematical laws" has received exactly the same attention as "universal mathematical piglets". I'd like to think you didn't make up the idea of universal mathematical laws. Can you give me the citation where you first heard of it?BarryR
September 12, 2010
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markf@163
The interesting question is do the axioms of integer arithmetic which we learn intuitively at primary school reflect some kind of metaphysical reality which alternative systems do not? Or do they just happen to be extremely useful?
Let me add a short clarification to that. I'm perfectly happy conceding a metaphysical reality to axiomatic systems that results in 2+2=4. However, I see no reason to think of this as the only metaphysical system out there (especially since a fair bit of my graduate career was spent learning about these other systems).BarryR
September 12, 2010
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StephanB@153
In keeping with that point, I gather that you have never heard of the law of limits with respect to calculus.
Do you remember why we use limits in calculus? It's because there's no smallest positive real number. The proof is pretty simple. By contradiction: let x be the smallest positive real number. x/2 < x, thus the contradiction. But math is all about choosing the axioms you want to work with, and so I can state as an axiom: there exists a smallest positive real number, and I'll call it an "infinitesimal". I can then construct calculus without all of those messy limits. My solutions will have a lot of infinitesimal left lying around at the end, but since they're arbitrarily small, I can disregard them. This is how Leibniz came up with calculus --- later formalizations preferred limits to infinitesimals, and that's probably what you learned. But the infinitesimal approach has been formalized as well, and is called non-standard analysis. It works, and all in all is probably a better way to teach calculus. As I understand your way of thinking, a smallest possible positive real number either exists or it doesn't. That's not how mathematicians think. They ask that question with regard to specific choices of axioms, and based their choice of axioms in part on how they can solve the particular problem they're working on.BarryR
September 12, 2010
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F/N 2: It is a little disquieting to see the same distractive fallacy of equivocation continuing. BR: I am very aware of how the glyph "+" may be given alternative meanings, both from mathematics and from the common use as the inclusive OR in digital logic. But the basic issue is not whether one may construct a set of axioms, hope they are coherent [a la Godel] and that they imply enough to be interesting and useful. Ordinary arithmetic and its extgension to school algebra are very useful, and reflective of commonly encountered reality. The reinterpretation of 1, 0 + and = in Boolean Algebra has significance for digital electronics and even reasoning in logic. But in all cases, one is very careful to distinguish contexts so one avoids errors of equivocation leading to gross contradiction. And, BTW, the act of joining the sets (*, *) and (*, *) to yield the set (*,*,*,*) is a very natural one, and to symbolise it as 2 + 2 = 4 is reasonable, meaningful and reflective of reality. To reason in a digital context that TRUE AND/OR TRUE is TRUE is a different and equally useful context, where 1 + 1 = 1, using 1 for TRUE and + for AND/OR (Vel, not Aut). To see the point that hose who object to your equivocation are making, consider a bottle bearing the string of symbols: GIFT In English, that's fun, but if that bottle came from Germany, watch out! In short, life and death may hinge on the context of symbols and being very caregul indeed not to be equivocal. And that has nothing to do with whether another set of axioms, for Graph Theory, may have utility or even map well to reality as we experience it, in some specialised circumstances; especially when we have to deal with networks of nodes and arcs connecting them in many modern circumstances. (BTW, does anyone say "Topology" much anymore?) The importance of contextual consistency and precision in terminology is underscored. GEM of TKI PS: For those who are wondering, GIFT in German, notoriously, means poison. Talk about a "False Friend" word!kairosfocus
September 12, 2010
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---vividbleau: "I think one could maintain that Lewis could have been a professor of philosphy and not a professor of philosphy or both simultaneously." Very good, vivid. According to our postmodernist friends, there are no laws that would force us to rule out that possibility.StephenB
September 12, 2010
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---BarryR: "When I played baseball in the backyard as a young child, each game was preceded by a ritual construction of the rules. The dogwood tree might be first base, the corner of the sidewalk third base, invisible men advance one base ahead the runner, etc." Apparently, you have been so steeped in postmodern subjectivism, that you cannot distinguish between socially constructed rules and objective laws. That is unfortunate. ---"Mathematicians are likewise free to choose their axioms to suit their fancy. Once the axioms are chosen, then they have to play by the rules (until the next paper)." Irrelevant. Mathematicians are not free to pick and choose which mathematical laws they will honor and which ones they will not. I have already listed six laws [there are many more] completely refuting your misguided claim that there are no laws. You simply ignore the refutation and continue on as sleek as ever. ---"See citations upthread for heuristics mathematicians use to evaluate their choice of axioms." Irrelevant as indicated in the preceding paragraph. ---"Do you have an example of an unchangeable law from graph theory? I’m not aware of any, but then you may know more about the topic than I do." Irrelevant. You could make your question relevant by showing how graph theory supercedes mathematical laws or reduces them to a status of not being laws. Perhaps you can provide a quote from one of your authors indicating that mathematics has no laws. [Good luck with that one].StephenB
September 12, 2010
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"Because I was free to disregard this universal rule, I was able to solve problems several orders of magnitude larger than what had been done using the discrete approach..." Why Barry, you must be the superman. Perhaps that explains your enthusiasm for Shaw. I remember our fifth grade teacher informing us one day, with a twinkle in his eye, that 2+2 could just as easily mean 3 or 5, by the rules of mathematics. Tell me, do you have a twinkle too? And Mark, you are also my hero. It's quite amazing to find one of our contemporaries inventing nominalism. And we thought time travel was impossible!allanius
September 12, 2010
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Looking over the 2+2=147 discussion I think it is worth restating what the issue is. I don't suppose that BarryR denies that given the axioms of integer arithmetic - incuding the definition of the "+" function - then 2+2=4. And these axioms are really useful for dealing with apples and countless other things. It is also possible to change the axioms (call it redefining the meaning of "+" if you like) so that 2+2=147 - which is practically useless in the real world. The interesting question is do the axioms of integer arithmetic which we learn intuitively at primary school reflect some kind of metaphysical reality which alternative systems do not? Or do they just happen to be extremely useful? No amount of sarcastic comments about bank cashiers is going to throw light on this question. Here is one approach. How do we learn that 2+2=4? Think back to primary school. Teachers do not reveal a metaphysical world of numbers to 5 year olds. They show them apples and such like, and give them techniques for counting them etc. Imagine trying to teach even the most intelligent child that there is this thing called a number, one of the numbers is 2, and there is a thing called addition, and when you add 2 and 2 you get another number called 4. And by the way these things don't exist in space or time - but the law that 2+2=4 is necessarily true for all space and time.markf
September 11, 2010
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#152 and #156 re C S Lewis I don't understand why you both repeated what I wrote? CS Lewis was a philosophy tutor for just under a year. That is a very long way from being a professor of philosophy as claimed in #146 - it doesn't require any original writing at all. He did indeed teach at Oxford for all those years but not philosophy and not as a professor.markf
September 11, 2010
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StephanB, Just for fun, I looked around a bit to see if anyone had defined exponents for graphs. I'm not seeing anything, so let's walk through what this might involve. You can define an unweighted, undirected graph as an adjacency matrix A s.t. the value at i,j is the number of edges from i to j. So we've got a matrix and that has a well-defined exponential function. But what if we have a weighted graph? If it's simple, then i,j can be set to the weight of the edge. (Can you transform a graph with weighted edges and verticies into an equivalent graph with just weighted edges? Yeah, I think so.) So raising this to a power is straightforward as well. But what if it's a directed graph? Now we're going to have to change our adjacency matrix to represent weight as well as direction. Do you order the verticies and use two matricies (one for edges going from high to low and the other for edges going from low to high)? Do you use a single matrix and give each vertex an alias? Do you create a matrix of vectors? You can do any of these things and then operate on the resulting matrix or matricies as usual, or you might get clever and come up with some else. And that will be your "Law of Exponents" for graphs. If this allows you to solve an interesting problem (or even raise an interesting problem), then your convention might be picked up and eventually make its way into textbooks. (Learning to understand what is "interesting" will occupy the first few years of your grad school career.)BarryR
September 11, 2010
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markf, From October 1924 until May 1925, Lewis served as philosophy tutor at University College during E.F. Carritt's absence on study leave for the year in America.Clive Hayden
September 11, 2010
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StephanB@153
Having studied higher math myself in a formal setting where bluffing doesn’t work, I am not buying your argument from authority.
Well you certainly shouldn't take it my authority.
I remember well enough the law of exponents and many other laws which admit of no exceptions.
You're familiar with exponents then, yes? Good. n^{2} is n-squared, right? n^{1/2} is the square root of n, right? n^{-1/2) is 1 divided by the square root of n, right? And you're familiar with i (or j, if you're a physics major) standing for the square root of -1, yes? And that's called an imaginary number. So what is i^{1/i}? If you wrote down what you considered to be the law of exponents, I don't think you'd be able to solve this problem (and it is solvable). But if we're allow to modify and extend the "Law of Exponents", then it's no problem. My reading group just finished this problem in Knuth this past week:
1.2.4.19: (Law of Inverses.) If n is relatively prime to m, there is an integer n' such that nn' is proportional to 1 modulo m. Prove this....
n'=n^{-1}, and now we're well on our way for defining exponents over modulo arithmetic. If you want to pick the "Law of Exponents for simple arithmetic over integers" as an axiom, that's fine --- just like Calvin and Hobbs above can decide to use "twelfth base" in baseball. But I'm not required to use that law --- even when I'm doing simple arithmetic --- and to my mind that doesn't make it much of a law.
So why on earth would I want to discuss your irrelevant foray into graph theory
Because it's your best shot at showing me I'm wrong. And because I thought we might have it as a shared mathematical language and we could elevate this discussion to axioms and theorem instead of discussing what you would prefer math to be like.BarryR
September 11, 2010
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StephanB@147
“Of course it’s undergirded by logical laws! When I use graph theory, I pick and chose which logical laws form that undergirding!”
Now you are saying that you want to choose from among the laws. Earlier, you argued that there ARE NO LAWS from which to choose.
When I played baseball in the backyard as a young child, each game was preceded by a ritual construction of the rules. The dogwood tree might be first base, the corner of the sidewalk third base, invisible men advance one base ahead the runner, etc. Once the game began, we were constrained to adhere to those rules. However, there were no rules constraining how we initially selected our rules. We were free to make the dogwood tree home plate, create additional bases that needed to be visited in arbitrary order, remove the strike count limit, etc. Mathematicians are likewise free to choose their axioms to suit their fancy. Once the axioms are chosen, then they have to play by the rules (until the next paper). See citations upthread for heuristics mathematicians use to evaluate their choice of axioms.
Just so you will know, definitions and assumptions are changeable; laws are not.
Do you have an example of an unchangeable law from graph theory? I'm not aware of any, but then you may know more about the topic than I do.BarryR
September 11, 2010
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