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wtf

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Everything posted by wtf

  1. In terms of nature, there are no regularities at all. There are no perfect circles, no straight lines, no triangles or spheres, no perfect symmetry. And in terms of abstract math, there are far more irregular and random objects than regular ones, if by regular we mean at the very least computable; that is, capable of being generated by an algorithm. Most functions are highly discontinuous, most real numbers are not ocmputable, and so forth. One shouldn't confuse the nice functions they meet in calculus class with all the wild functions that are out there. One should also not allow oneself to be confused by the apparent regularity of physical law. What we mean by physical law in this context is NOT the true nature of the universe; if there even is such a thing. Rather, by physical law we mean the historically contingent human-created scientific theories of the universe. All such theories are at best clever approximations. Our best physical theories are good to 12 decimal places or so. That's terrific as physical theories go. But they're not exactly what nature does. The apparent regularity of nature may arguably be telling us more about our own minds than it does about nature.
  2. I feel that I appear to have been riding a hobby horse; when my intention was only to express my opinion, and then reply to my mentions. In fact why would my opinion, expressed once a couple of weeks ago, prevent a meaningful discussion of AI? That right there is a good question. Why don't people discuss AI? That's the point of the thread and it's the reason why I'm here. My remark about emergence was intended to be an offhand expression of a minority opinion that I happen to hold. People started pushing back and I've been replying. I'm not stopping anyone from talking about AI and I wish someone would. I've been quite surprised at the reaction to my opinion. In terms of the subject of the thread it's not important to me at all. And I said exactly that about 8 posts ago if I recall. I simply expressed an opinion. I always reply to mentions.The more I reply, the more people think I'm invested in the topic. I'm not. I have an opinion, that is all. Here is my last word on the subject. I looked around for criticism of emergence and found two links that made me sufficiently happy to feel that I've at least made my point to myself, if nobody else. 1) From this SEP article I found two examples. https://plato.stanford.edu/entries/properties-emergent/#ObjEme After A technical argument which I didn't try to follow, the article says: So at least one professional philosopher is willing to use the word "incoherent." It's not just an over-the-top word invented by me. I am not the only person who finds emergence incoherent. Me for my amateur philosophy reasons, and Kim for his professional philosophy reasons. And note the word epiphenomenal. It's a good description. Something that's there, but not essential because the thing in question doesn't need it. SEP goes on to discuss another critic. Again as with Kim we find that emergence is regarded as an epiphenomenon. That is, something that shows up whenever quarks turn into elephants, but that would make no difference if it didn't show up. Quarks turn into elephants whether you call it emergence or not. I think that's the point being made by calling emergence an epiphenomenon. 2) Eliezer Yudkowsky. Of all people. Do readers know who he is? According to his Wiki entry he is: As I understand it, some consider him a genious and others not so much. I've read him a little but never been much of a fan. But he wrote this essay ... it's literally word-for-word what I've been thinking. I could quote it but I'd really have to quote the whole article. I hope people will read it. This essay is so uncannily like my exact thoughts on the matter, that it's not out of the realm of possibility that I actually read this many years ago and that's where I got my own ideas. I really can't say. I hope people will read it. It's as good a presentation of my ideas as I wish I could have written. I don't expect to convince anyone but at least I'm not alone. The Futitliy of Emergence Here are a couple of quotes. and So anyway to sum all this up, the set of professional philosophers who agree with me is nonempty. And for what it's worth, Eliazer Yudkowsky agrees with me. I'll take my agreement where I find it. I will now stop talking about emergence. I haven't said anything new for quite a while and the Yudkowsky article expresses my thoughts perfectly. "Your curiosity feels sated, but it has not been fed." THAT is what I'm getting at. Emergence is an intellectual snack made of empty calories.
  3. Agreed. It tells us nothing but it is a shorthand for when a thing has properties that are not at all obvious from the properties of its components. I'm perfectly fine with that. I just don't think it's meaningful or helpful when applied to consciousness. Or much else for that matter. > It's a convenient way of facilitating discussion, however it only works if the all the participants understand the language. You've decided that calling something an animal tells us nothing. Well a horse is a lot different than the organic molecules that it's made of, so that's emergence. Ok fine. But if I want to know how horses work, I read a textbook on biology. I think we're in agreement. The word emergence is a classifier but not much of an explainer. And besides, I can barely think of anything that's NOT emergent. But the debate isn't about whether horses emerge from molecules, which they clearly do. The problem is that emergence is so often used to short-circuit discussion about consciousness. Someone above said that emergence can replace the idea of the soul. I don't see how it's any different at all with regard to consciousness. Brain goo is somehow self-aware. That's a mystery. You can call it emergence or you can say God imbued Man with a soul. Neither seems like a satisfactory explanation to me. I'd rather read a book on neuroscience. Not that neuroscience has a clue where self-awareness comes from. But at least it's not burying the question under a single magic word like soul or emergence.
  4. Ah. But this is false. We don't need emergence to explain how molecules form. All we need is physics and chemistry. There seems to be an ongoing theme in this thread that emergence is just a magic word to label whatever we don't understand. Consciousness arises in brain goo, which is a mystery. So we call it emergent. But we don't know any more by calling it emergence than we do calling it a mystery. Or calling it soul for that matter. We have science to explain how oxy and hydro, which aren't wet, become water, which is. Once it was a mystery, but now we understand it. Is it emergent? Sure, the word fits. But it doesn't explain anything, whereas physics and chemistry do. I'm really struck by your statement that in the absence of emergence there would be no meaningful structures. But the concept of emergence is only a label. Meaningful structures did emerge. But to the extent that we understand the process of structure formation, our understanding is due to science. Not to labelling things "emergent."
  5. I couldn't explain any more than I have in my last half dozen or so posts. Emergence tells me nothing I didn't know before. > What the word 'emergent' means is clear: that a system has properties that its parts do not have. Ok, so water, which is wet, emerges from oxygen and hydrogen, neither of which are wet. So that's emergence. But that doesn't tell me anything. What DOES tell me something is reading a book on chemistry, which explains the physical mechanism by which oxy and hydro combine to make water. But how does emergence replace soul? If I said that God puts a soul into your brain goo, you'd call that superstition or at best Cartesian dualism. But if you say that "emergence" puts consciousness into brain goo, you say that's much much better. I say it's no better at all. You're replacing one word for a mystery with another. What's the difference between soul and emergence?
  6. It may be true, but it's not useful. I keep repeating this. You say consciousness emerged from inorganic matter, but that's not proven. You have no actual evidence for that. But say I agree that consciousness "emerged" from inorganic matter. What do I know now that I didn't know before? Does consciousness emerge from rocks? Do I now know that? No. If I say water "emerges" from oxygen and hydrogen, I know nothing. I'd be better off reading a book on chemistry, which would elucidate the actual process by which oxy and hydro become water and doesn't use the word emergence. You can call it anything you like but you aren't telling me anything. It doesn't tell me anything. I still don't know what consciousness is or how it arises. Besides, consciousness is a terrible example because we don't even KNOW what it emerges FROM. Does consciousness emerge from brain goo? Or does consciousness emerge from computation that only incidentally happens to be implemented in brain goo, but might possibly be implemented in computer chips? The use of emergence in this context is doubly incoherent because its proponents don't even know for sure what consciousness emerges from.
  7. I have no objection to the word itself, or the concept. I disagree with those who use it as an explanation or an answer. People will say that consciousness is emergent, as if they've told me something. I don't feel that they've told me anything. I don't dislike the word. I just don't understand why some people find the idea so important or meaningful.
  8. Ok. Emergence is a classifier. Phenomenon X is or isn't emergent, depending on how you define it. But either way, it tells you nothing. Water emerges from oxy and hydro. Tables emerge from wood. Consciousness emerges from brain goo (or is it computation that just happens to be, but doesn't need to be, implemented in brain goo?) In every case we can discuss and argue whether or not the given thing is emergent from other things. But in NO case can we say that it adds clarity, or insight, or understanding. If consciousness emerges from brain goo (or substrate-independent computation) we have no additional insight or understanding that we didn't have before. So if you want to say that X emerges from Y, it's fine with me. But what does it tell us that we didn't already know before? Someone used the term emergence. I find the term incoherent.
  9. On the subject of emergence, I'll have more to say later and I plan to directly address your point. Between the post in my head and the post I can write might take a day or two. I'll get to it, I did actually figure something out. Right now I just want to talk about 2 + 3. I've been thinking about your 2 + 3 example and I want to explain that first, I understand what you mean; and second, that I think arithmetic is absolutely the wrong metaphor and moreover is bad because it just confuses people as to what numbers are. The reason I can't get with your arithmetic metaphor is simply that I have a math background. The symbology "2 + 3 = 6" is not to be overloaded vague or philosophical or ignorant reasons. You see a lot of the latter in advertising. Deliberate bad arithmetic to draw in the rubes. I actively oppose all such usages because they only serve to further confuse an innumerate public. Numbers are numbers. That's what's important about them. They don't represent anything at all. And the arithmetic of the small positive integers is not to be taken in vain. It's often used philosophically as an example of something that must be true in the universe even in the absence of sentient beings. It's commonly held even in philosophy that the arithmetic of the small positive integers has a special status among all abstract ideas. We do believe it must be true in all possible worlds. So I get the point you're trying to make. I just object to the arithmetical metaphor. That said, I do agree that perhaps 2 grams of X plus 3 grams of Y may output 6 grams of Z. But that can't be. Mass has to be conserved unless you're doing some kind of relativistic thing. In terms of chemistry, that can't happen. But that's not what you mean. You mean to say that substance Z, made from substance X and Y, has qualities and properties that are in no way inherent in X and Y; and that are in fact surprising in that regard. That surprise factor is what we call emergence. So we mix oxygen and hydrogen and get water, which is wet. That's emergence. Again from a math perspective I object to using quantity to try to say something about quality. I object to that use of the natural numbers. 6 is more than five. Water is different than oxygen and hydrogen. It's not "more." People often misuse numbers that way. Increasing the aggregate amount of numeracy in the world is one of my life missions. I can therefore in no way accept your arithmetic metaphor; even if I get what you're saying. To sum up: 1) I understand what you're trying to say; and 2) Arithmetic on the small positive integers is absolutely the wrong metaphor; and I object to it. You want to metaphor-ize a qualitative difference. Your use of arithmetic can only annoy people who know math; and further confuse those who don't. More on emergence later. I do have a little thesis in mind, it will take me some time to write down.
  10. What's the problem with 2 + 3 = 6? What on earth are you talking about? That's just a falsehood. Obvious to me? No. I honestly can not imagine what your point could be. > We can only say they emerge. Ok. So after you tell me that, what do I know that I didn't know before? You might as well say it's due to axolotls or frisbees.
  11. In other words, false? Of course it did. As a chair emerges from atoms, my pancreas emerges from electrons, and consciousness arguably (but not yet conclusively) emerges from brain goo. And once you've told me that, in each case I know nothing I didn't already know before you told me.
  12. Was this for me? I sometimes don't read posts unless I get a little red mention rectangle when I come to the site. If there were mathematics to describe how consciousness comes out of brain goo, we wouldn't need the name emergence. Just like when we add hydrogen to oxygen in a lab, apply heat, and end up with water. We don't call that emergence, we call it chemistry. A chemist can describe the exact mechanism by which it happens. Likewise when we take pieces of wood and create a chair, we don't call it emergence, we call it carpentry and we know exactly how the process works. It seems that emergence is just a placeholder for processes that we don't understand. And after we call these processes emergent, we don't understand them any more than we did before. I am not sure I followed the example of addition. I didn't question the usefulness of addition. But addition of like quantities (centimeters in your example) is not an example of emergence. Consciousness somehow arising from brain goo is called an example of emergence, but that's a long way from addition of small integers.
  13. Computers (hardware + software) can only do exactly what they're told. If they're told to search a corpus of data, build a neural net with weighted nodes based on the data, and produce outputs based on the weightings, then that's exactly what it's told to do. It is not magic, it's deterministic coding. [And even with randomness, nondeterministic automata don't compute anything new]. I'm not sure if the emergence remark is sarcastic or not. I'm not talking about emergence anymore. I've heard about it and read about it for years and my opinion is, rightly or wrongly, the product of some thought. Yes but that's true of EVERYTHING. My chair is more than a bunch of atoms. My liver is more than a bunch of molecules. The Mormon Tabernacle Choir is more than a bunch of singers. So yes, emergence is true. I don't deny that. It's just useless because it tells us nothing. Practically everything is more than the sum of its parts.
  14. I'll stop talking about emergence. My job now is to read more. I'm perfectly well aware that a lot of smart people find emergence meaningful. I don't. Hydrogen's not wet and oxygen's not wet, but water is wet. Ok. I don't find that interesting or meaningful. Atoms don't have chair-ness in them but my chair's made of atoms. What of it. But when it comes to consciousness, it's even worse. If you tell me that consciousness is an emergent property, that tells me even LESS than telling me wetness emerges from hydrogen and oxygen. Because with consciousness, you can't even tell me WHAT it emerges from. Does consciousness emerge from brain goo? Or from computations that may be implemented in any suitable substrate? Nobody knows. So calling consciousness emergent is doubly meaningless. First because emergence in general is meaningless (in my opinion) and secondly because nobody can say with authority what consciousness even emerges from. Having now gotten in one more remark about emergence I will go do some more reading on the topic. Perhaps I'll learn something. This is not true. The most sophisticated deep-learning AI in existence -- let's say AlphaZero, to be specific -- is a physical implementation of a Turing machine running on perfectly conventional hardware. It has no more computing power than any Turing machine. We could in theory implement AlphaZero using a human being equipped with nothing more than a pencil and a sufficiently long strip of paper on which a sequence of cells have been marked. This is a fact of computer science. I was going to write a long explanation of this point but perhaps I'll leave it at this and try to answer any specific questions. Deep learning and in general machine learning (ML) techniques are very clever ways to organize a Turing machine. They don't do anything new. They are based on analyzing a corpus of data, assigning weights to nodes, and making decisions based on the weightings of the nodes. It's perfectly conventional programming done with conventional programming languages and running on conventional hardware. These programs are deterministic. Even if they are equipped with "randomness" via a pseudo-random number generator or even a physically-based RNG, it's well-known that introducing randomness into computations does not increase their computational power. A Turing machine can be programmed to simply execute ALL possible logic paths at any branch point to simulate the element of randomness. It's true that ML algorithms do "build their own neural network" by continually refining the weight functions by which they evaluate their data. But they are programmed to do so, and the programming is perfectly conventional. In short, contemporary AI's don't go beyond Turing machines. They're simply very clever applications of Turing machines. And not all that new; abstract neurons were conceived in the 1940's. AI's do exactly what they're told. We know of no other way to implement computation.
  15. I've read all the replies to my posts ... that is, all the replies that generated a mention. I did not read replies that didn't generate a mention. If I missed a cogent explanation of emergence, I'd be grateful for the pointer to whatever i may have missed. If I study everything known about a given scientific topic; then you tell me in addition that it's "emergent"; what do I know then that I didn't know before you told me that? ps -- David Chalmers on emergence. He then goes on to more detailed analysis. He seems in principle to have some of the same misgivings I do. Perhaps he's ignorant and raging. http://consc.net/notes/emergence.html
  16. Then why can't anyone give a decent defense or explanation of the emergence? I've asked simple questions. Say you know everything there is to know about the aurora. If I tell you in addition that it's "emergent," what do you know now that you didn't know before? I say emergence is a vacuous (but trendy) concept. That doesn't necessarily make me ignorant. Another interpretation would be that I've read quite a lot about emergence and it sounds like fashionable nonsense to me. Did you get anything from the Feynman story? I find emergence to be along the same lines. A word that is supposed to explain everything but explains nothing.
  17. What causes life? "Tunawaddles." Adds no information. What causes life? "Emergence." Adds no information. Take the earlier example of the aurora borealis. You could study physics to learn why charged particles give off light. You could study atmospheric science to understand everything we know about the aurora. Then if I tell you that the aurora is "emergent," you haven't got any more information or understanding than you did before. Feynman made this point in one of his books. He told the story about how he got roped into serving on a school textbook committee. The book said, what cause such-and-so? Energy. What causes this-and-that? Energy. And Feynman exploded. He said just using the word "energy" to explain things was meaningless. Same with emergence. What's the aurora? Emergence. What's life? Emergence. What's consciousness? Emergence. In every case you know nothing that you didn't already know before someone stuck a silly label on it. I say emergence is a vacuous concept. It has no clear meaning. It explains nothing. It adds no clarity or insight. ps Here is the Feynman story. Or, I'd add, "emergence" makes it move. Feynman added: https://fs.blog/2016/07/richard-feynman-teaching-math-kids/
  18. My seeming rage? Is there a language issue? I'm poking on the concept of emergence. I think it's quite murky. There are philosophers who agree with me. I guess I can't engage in further discussion till I get clarity here. If I'm upsetting you somehow I'll stop replying to you. I don't care if you call consciousness emergent. I'm trying to get you to explain to me what that adds to anyone's understanding of consciousness. If you can't tell me that, then perhaps emergence isn't a useful concept. Do you think molecules are an emergent property of atoms? I'm trying to get at your definition here. Do you see emergence as a component relationship? Or something else? Well that's enough rage for now I guess.
  19. Hi, thanks for the links. I will take a look at those. I clicked on your first link expecting to see something about Tononi's idea of integrated information theory but evidently this is something else. I don't think there's a mathematical theory of consciousness. And even if there were, it wouldn't necessarily be true or false; just as there's a mathematical theory of Euclidean geometry whose truth or falsity is a matter for physicists and not mathematicians. Mathematics is no longer a path to truth. Between the 1840's and the 1930's Riemann, Einstein, and Gödel busted up that idea pretty well. But I'll take a look at the article. The second looks interesting too. Axioms for consciousness. Well why not. Again they wouldn't be true or false, but they might be interesting. I think when it comes to consciousness perhaps we're hoping for a "WHY" as well as a "WHAT." An explanatory theory as opposed to merely a descriptive one. But science doesn't do metaphysics. It only describes the outcome of experiments. Science is very limited that way. This was in response to "But with consciousness, you have told me no such thing. You do NOT have ANY theory of the mechanism by which consciousness "emerges" from .... actually you didn't tell me what it emerges from! The brain, the body, a bunch of circuits ... you have not told me. " Well you go ahead and take a run at it please. I'm sure you can explicate a lot of contemporary neuroscience, some of which I'm familiar with. But none of it explains consciousness. If you think it does, do feel free to explain it to me. But the original point I believe is the concept of emergence. If you have a specific technical path from X to Y, then Y isn't emergent. In other words molecules aren't emergent from atoms. Emergence implies something BEYOND a mere component relation. And that beyondness is what I'm highly doubtful of. This was in response to my asking: "What can "objectively sentient" mean? Is your next door neighbor objectively sentient? How do you know?" I can't agree with what you wrote. I can write a program that accesses a database of yes/no questions and their answers by a large number of people (Is killing bad? Is apple pie good? Are fish mammals? etc.) and then my program would simply output the majority answer to whatever question is asked of me. I wouldn't need any special AI programming techniques. It's a simply database lookup. Just a few lines of code once you have a database of common questions and the average answer by a large sample of people. I hope you can do better than that for a definition of "objective sentience." If sentience is simply spouting conventional wisdom, a news bot that reads you today's New York Times in a mechanical voice would be sentient. I'll be charitable and say that I understand you to be trying to express the idea of empathy. We all know that in general, killing is bad. (Then again there's war, there's capital punishment, there's the defense of your own life or that of your loved ones, etc.) You think if a machine says killing is bad it's because the machine has empathy or sentience. But you can just as well program a machine to output "Killing is bad" in response to the question, "Is killing generally bad?" and that is not sentience. That's just a lookup table.
  20. If you tell me the northern lights are emergent I don't know anything about them that I didn't know before ... until you add information. If you're going to tell me something is emergent, you have to tell me HOW it's emergent. And in this case you did. You said: "the way particles from the sun interact with our magnetic field and atmosphere." Ok! An atmospheric scientist could probably drill that down to the molecule for me; and then a physicist could drill it down to the quarks, the way Feynman explained how light interacts with matter. But with consciousness, you have told me no such thing. You do NOT have ANY theory of the mechanism by which consciousness "emerges" from .... actually you didn't tell me what it emerges from! The brain, the body, a bunch of circuits ... you have not told me. So your claim that consciousness is emergent is missing two things: 1) What does it emerge from? And 2) How. Now that's funny. You are defending the thesis that consciousness is an emergent property (of something, you haven't said what). And now you want ME to define consciousness for you. Surely you must have a definition in mind yourself, in order to make such a definitive pronouncement with such certainty that "consciousness is an emergent property." I had to use caps because it was vital for me to communicate that I am much less interested in the answers to those questions; than I am in the fact that the idea of "emergence" doesn't help me answer them. I'm attacking the concept of emergence. I honestly do not believe in it. I have read about it. I don't find the idea compelling. It gives a name to something but imparts no understanding. I hope you took to heart my example of the northern lights. You told me it was emergent. You then told me WHAT it was emergent from; and you even told me HOW that emergence takes place. You have not done those two things with consciousness.
  21. NO THAT IS NOT THE POINT!! THAT IS NOT THE POINT!! I"M NOT RAISING THE QUESTIONS TO ASK THEM. I'm pointing out simply that: 1) if I have those questions in front of me; and 2) I fully embrace and accept the statement that "Consciousness is an emergent phenomenon," THEN I have NO MORE INFORMATION NOW as to what the answers are, than I had before accepting the statement in (2). That's my point. Yes these are great thought questions. But I do not care about them right now. I am pointing out that if I assume that "Consciousness is emergent" or I DON'T assume that; it makes no difference in helping me answer those questions. The statement: "Consciousness is emergent" conveys no information. This is my point. I know I posed some good thought questions. I'd be frustrated if people started answering them! I am laser focused on the concept of emergence. My thesis is that it doesn't mean anything, in the sense that it adds no new information to discussions of consciousness. So if you have a thought or two on the questions, please throw me a bone and try to explain to me why anyone thinks "emergence" actually conveys any new information.
  22. Perhaps, to help me understand, you can explain what issues it illuminates for you. Let us stipulate that "consciousness is an emergent phenomenon." For sake of this conversation I accept that proposition. Can you tell me: * What does it emerge from? * Must it emerge from something living? Or may it possibly emerge from particular arrangements of inanimate objects? * If so, might it emerge from particular arrangements of rocks? How about electrical circuits? What is the difference? * If, as we are told, consciousness is a computation; then by the laws of computation it computes the same thing whether it is made of computer circuits or dominoes. (Google the domino computer). * If we implement an AI in a computer made of dominoes, as can perfectly well be done, can consciousness emerge from such an apparatus? * Do you think consciousness can emerge from Microsoft Windows, which I believe consists of around 100 million lines o code? * Do you think consciousness can emerge from the global supply chain? The network of computers and logistics systems that manage all the ships and planes and manufacturing facilities that get raw materials from one country to component factories in another to assembly facilities in a third and to the retail shelves in a fourth? The global supply chain is by far the most complex system we have, far more inscrutable and unknowable than the fanciest AI. People just don't think about it because we don't think about how stuff gets to retail outlets. But it's pretty complicated. Now my point isn't only to stimulate discussion about these questions and others like them. My point is that saying, "Consciousness is an emergent property," does not give me a clue about how to answer ANY of them. It tells me nothing. That is my point. But you say it tells you something. If you can explain this to me it will help me very much in understanding why SO MANY seemingly reasonable people say that "Consciousness is an emergent property" is meaningful to them. I hear this a lot. I just don't understand it.
  23. I don't find emergence to be a helpful concept. Suppose consciousness is emergent. What did you just explain to me? Nothing at all. I still don't know what it emerges from. A particular configuration of molecules? Or must they be organic molecules, ie living things? Could sentience emerge from a digital computer? Emergence is a magic word that makes people think they understand something when they've actually understood nothing. If sentience is an emergent property, what does it emerge from and what doesn't it emerge from? Could it emerge from the 100 million lines of code making up Microsoft Windows? Could it emerge from the global supply chain? How about Skynet? Could it emerge from a video game? Could it emerge from the code running Ms. Pacman? In which case did I murder a sentient being every time I ran out of quarters back in the day when I played video games in bars? Emergence sheds no light whatsoever on any of these questions.
  24. No, there are many such functions. |x| is not one of them.
  25. Ok. You look perfectly sensible doubling down on your error. You look like a stud. A God. A man among boys. Happy now? Studiot you are the one not sticking to the math. I posted the correct answer. You're still trying to defend your incorrect point. What math should I stick to? I first noted that the "obvious" answer, which more than one person proposed, is wrong. I then looked up the right answer, which I was not able to work out for myself. I'm not claiming any special virtue here. So what more do you want? You're unhappy that I said you look silly? Ok you look terribly clever. I would also argue that "You look silly" is not an ad hominem. If I say you ARE silly that would be an ad hominem. But you're not always silly. Sometimes you display an impressive grasp of engineering math. Today, you are insisting you're right when it's perfectly clear that you're wrong. But if you want me to apologize for calling that behavior silly, I apologize.
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