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| What The Tortoise Said to Achilles. | |
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| Topic Started: 2 May 2012, 10:12 AM (931 Views) | |
| fdrake | 2 May 2012, 10:12 AM Post #1 |
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Lewis Carroll is awesome. Achilles and the Tortoise are walking down a street one day, and Achilles, having studied some basic geometry, decided to air his thoughts on it to his good friend. Achilles: Assume line A is parallel to line B. Also that line B is parallel to line C. Therefore, line A is parallel to line C, is it not? Tortoise: It would certainly seem so. But what about A being parallel to B and B being parallel to C allows you to infer that A is also parallel to C? Achilles: Well, let me rephrase, if line A is parallel to line B, and line B is parallel to line C, and if line A is parallel to line B and line B is parallel to line C then line A is parallel to line C, then line A is parallel to line C. Tortoise: It would certainly seem so, again, but why is it so that if line A is parallel to line B and line B is parallel to line C then line A is parallel to line C along with A being parallel to B and B's being parallel to C allow you to conclude that A is indeed parallel to C? I can definitely visualize the lines, but I cannot follow your proof. Achilles: I don't know how to rephrase, every time I do it you'll just plug my statement into this formula: suppose that I've stated that some propositions 1 and 2 imply some proposition X, you may tabulate it like this, right? *he draws a few symbols in the sand*. 1. (1) 2. (2). These were my starting premises, I assumed: 3. (1) & (2) imply X. So you questioned this, and I was forced to include this as a premise thusly: 4. (1) & (2) & (3) imply X You then questioned me again, and I was then made to include: 5. (1) & (2) & (3) & (4) imply X As a premise... I'm willing to bet if you were bored you would repeat the process, wouldn't you? Tortoise: What leads you to that conclusion? Achilles: Well you did it before. Tortoise: So what lets you conclude that if I was bored I would repeat the process if I did it before? Achilles: So it seems you can apply the above scheme to doubt anything. Tortoise: I never said I doubted it, only that your reasoning did not demonstrate it. I take Achilles' conversation with the Tortoise to mean that there can be no sufficient conditions to lead us to conclude thatf P such that these sufficient conditions are invariant of all possible interpretations. That is to say; whenever we speak, we must share a parlance and an some trends in outlook, and these codetermine which inferences we hold firm. Hence: "theism isn't rational" is often little more than an appeal to the familiar, perhaps you atheists should be more open minded? |
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A stranger has come To share my room in the house not right in the head. | |
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| jayfoxpox | 2 May 2012, 11:08 AM Post #2 |
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Alright so the first example is pretty much "if a is b and b is c then a is c." then the second one seems so different I'm not even sure if it's the same logic. Looks like (A & B) & C..............& n therefore X. The way I typed it is a little flawed , but pretty much A and B equals X and adding anymore conjunctions also equals X . Maybe the message is , by trying to rephrase your logic you run the risk of strawmaning yourself? |
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| Categories+Sheaves | 2 May 2012, 02:15 PM Post #3 |
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Pedantic Curmudgeon
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Oh man, I remember coming across this in GEB (Hofstadter)... Does this mean you're currently reading said book, or are you such a big Lewis Carroll fan that you're getting it straight from the source? |
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So these philosophers were all like, "That Kant apply universally!" And then these mathematicians were all like, "Oh yes it Kan!" Daily Dose of Zizek Invictus Fancypants Math Rants
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| fdrake | 3 May 2012, 06:32 AM Post #4 |
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I've read GEB! And the original article. I wanted to bring it up since it seemed related to Taelok's posting on how Christians go about the business of having faith. Sort of! The idea in the dialogue isn't that (P->Q) -> (P&R -> Q), that is to say if you know something and you know that implies something else, knowing additional stuff still at least lets you conclude the original conclusion. It's this: say you have a list of premises to an argument, call em {1,2,3,4,...,n}, and from these you want to conclude some proposition X. Then it seems Tortoise can assert: "{1,2,3,4,...,n} implies X was another premise.". Consider that this wasn't the case. That is to say that "X is false" is compatible with {1,2,3,4,...,n}, then "{1,2,3,4,...,n} implies X" would have to be included in the premises of the argument. Achilles then labels this implication as premise (n+1) and adds it to the list: {1,2,3,4,...,n,n+1} implies X. Which yields Tortoise to repeat the line of questioning before. Since he can do that indefinitely, it follows (by induction) that no finite list of premises to an argument can ever provide a grounding for any conclusion. Some "magic" has to happen in order for us to infer. Presuming that this "magic" is one way rather than another [like atheists charging Christians with "it's not rational!"], is little more than an appeal to the familiar, IE which standards atheists tend to judge arguments by; namely some nebulous sense of scientific rationality. This is the "magic"; alien, by the above, to what may be rigorously derived. Edited by fdrake, 3 May 2012, 06:47 AM.
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A stranger has come To share my room in the house not right in the head. | |
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| Ardat | 3 May 2012, 08:19 AM Post #5 |
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Stupid conclusion. All you have to do to shut that fucking tortoise up is to assume transitivity. It can then question the inference all it wants, but I defined a system where it's simply true. The difference is, transitivity is a reasonable axiom. What's called "rational" by most atheist is a reasonable set of propositions and relationships between those propositions, such as "With carbon dating, we can see the Earth isn't 6000 fucking years old". They aren't rigorous, but they certainly have been proven effective to model reality, which justifies their perceived reasonability. Being completely rigorous at all times is simply an inefficient way to apprehend reality at this point. Are you implying theists are in any way more "rational"? Edited by Ardat, 3 May 2012, 10:55 AM.
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| What's more fun than fighting crime? | |
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| jayfoxpox | 3 May 2012, 08:23 AM Post #6 |
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ah yea I see, induction isn't logically valid ( certain) , but at best provides a strong argument. So in essence , both sides are guilty of a logically fallacy. |
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| fdrake | 3 May 2012, 08:30 AM Post #7 |
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@ardat:
Edit: "Ah right, so "transitivity" is another premise in the argument. Restatement of the problem, surely." I realized that the Tortoise also assumes transitivity by using induction (if he wants to conclude it for all arguments, anyway, rather than doubting some instances of arguments, which makes his method more watertight), so transitivity's fine by assumption of the problem. However, you still need to include the particular implications you use as premises in the argument. So while "If A=>B, and B=>C then A=>C" is true. The tortoise doesn't need to doubt transitivity: If A is parallel to B, and B is parallel to C, then A is parallel to C presumes: "A is parallel to B and B is parallel to C implies A is parallel to C", which isn't a substitution instance of "If A=>B, and B=>C then A=>C" since being parallel is assumed to be transitive, rather than implication. So rather than doubting the transitive property of implication, the tortoise always questions whether the implication "these premises => the wanted conclusion" is true or not. If it's not true, then the argument fails, if it is true, it must be included as a premise. Edit2: Now I'm really not sure whether transitivity is required for mathematical induction. Help me out categories? It seems that mathematical induction [not transfinite past the limit ordinal w] makes a statement about countable sets of propositions and transitivity of implication doesn't require this presumption. Similarly P(k)=>P(k+1) is pairwise implication, nowhere in the induction schema is it required to have ["P(0)=>P(1)" & "P(1)=>P(2)"] => "P(0)=>P(2)", only that [P(0) & for all k P(k)=>P(k+1)] => for all k P(k).
So where does this reasonability come from?
No. It might also be the case that they're "less rational" by usual metrics, but my point is by no means does this make what they believe in unjustified without an appeal to what is familiar [IE "reasonable"] to the atheist. Mathematical induction is logically valid, click the link. Edited by fdrake, 3 May 2012, 09:11 AM.
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A stranger has come To share my room in the house not right in the head. | |
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| jayfoxpox | 3 May 2012, 08:50 AM Post #8 |
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sorry what link? And when I say logically valid I mean as in knowing for certain. Like if there were 100 apples , then you grabbed 99 red apples. through induction it's highly probable the next apple is red , but it could be green , yellow . Strong argument's are not logically valid , but they are logically strong , which are probable and still a good argument. Weak arguments are also not valid and are improbable , but still possible , but a bad argument. to clarify on 4:44 If this is wrong then I want a refund since I bought an online course from him lol. |
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| fdrake | 3 May 2012, 08:54 AM Post #9 |
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This one. |
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A stranger has come To share my room in the house not right in the head. | |
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| Ardat | 3 May 2012, 10:54 AM Post #10 |
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Well sure, it's the transitivity of being parallel that needs to be assumed. I still fail to see how it's unreasonable, 'cept maybe in some fucked up geometries. Probably accurate, but Categories will certainly be of better help than I am here. You receive packages of information, interpret them, try to connect the dots to establish a beliefs system (where the logic is itself predicated by what you see as "reasonable"), and then use that system to establish predictions. If your predictions are systematically unverified, your system is a pile of shit and you can start over. If they are partly accurate, then you try to spot what's wrong by reviewing all the assumptions you made. It's an algorithmic process and in which, indeed, "reasonable" is akin to "familiar". But how is this a problem? You're being unfair. Atheists that solely appeal to what's familiar to them but insist on being militant anyway are commonly know as obnoxious douchebags. Plenty of them appeal to fields of knowledge that aren't contingent upon the premise of atheism, and that's what justifies the "more rational" prerogative. It depends where you work, doesn't it? |
| What's more fun than fighting crime? | |
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| Categories+Sheaves | 3 May 2012, 03:25 PM Post #11 |
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Pedantic Curmudgeon
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Asking me for a proof without using logical transitivity... you serve me a triple espresso, and then you ask me to sit still? (Or maybe this is more like the demand "stop thinking!") These demands of yours... Induction strongly suggests that it runs on nested transitivity... The [P(0) & for all k P(k)=>P(k+1)] => for all k P(k). is simply a statement of what is necessary to prove something via induction, not a statement about how the induction argument actually functions... as you mentioned, the 'naive' approach to induction strongly suggests this sort of schema: Spoiler: click to toggle But here is (I think) a nicer argument for why it need not rely so heavily on transitivity (this is the argument that we use on the transfinite ordinals!). Let O be a well-ordered set. If we replace [P(0)] & [P(k) -> P(k+1)] with the 'strong induction' statement used to perform induction on the transfinites, "If P(k) for all k < n, then P(n)" (to do this translation on N: drop an argument about how every nonzero element of O is the successor to something) you get the result that "{n in O:~P(n)} is empty" right away (by using well-ordering). This obviously isn't tortoise-proof (and it's just a sketch, anyway). But the fact that induction has to be taken as an axiom (#9 on Wiki's page for PA) should mean this question sits on some rocky terrain... Edited by Categories+Sheaves, 3 May 2012, 03:42 PM.
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So these philosophers were all like, "That Kant apply universally!" And then these mathematicians were all like, "Oh yes it Kan!" Daily Dose of Zizek Invictus Fancypants Math Rants
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| fdrake | 3 May 2012, 03:29 PM Post #12 |
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I'm not saying it's unreasonable. I'm saying it doesn't have to be given, assent may always be denied to this.
So what makes the theist's claims that atheists are being misunderstanding or unreasonable bunk? Whether a view is unreasonable, as you highlighted with your assumptions, is decided by what you hold to be the case prior to the conversation. Why's "God exists" not a vindicated "starting point" for discourse whereas "being parallel is a transitive relation" is? Rationality seems to fall from the sky, in your view, rather than God's wrath. |
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A stranger has come To share my room in the house not right in the head. | |
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| Ardat | 3 May 2012, 04:57 PM Post #13 |
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Who's strawmanning now? Do you feel like the statement "God exists" is just as reasonable as "being parallel is a transitive relation"? Reasonability is derived from what's already been established to accurately model reality, which is why borrowing results from various fields of knowledge not contingent upon atheism to give credence to its premise serve as a good argument. The reasonability of a claim is therefore mostly a function of how the predictions derived from it fit experimentation. You also have to consider the level your claim acts on. God can't be actively observed in its entirety, while a line is a much simpler object. That a claim is reasonable heavily depends on the tools you possess to see what it reflects upon. So no, rationality in my view doesn't fall from the sky, unless you define sky as thousands of years of technological development and millions of experimental results. Edited by Ardat, 3 May 2012, 05:06 PM.
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| What's more fun than fighting crime? | |
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| Dolerbom | 3 May 2012, 05:02 PM Post #14 |
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Can we stop having these same rehashed threads just with different wording and/or examples/experiments? |
| I heard you like short sigs... so i made this. | |
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| Ardat | 3 May 2012, 05:05 PM Post #15 |
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Oh I'm sorry, are we bothering you? |
| What's more fun than fighting crime? | |
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| fdrake | 3 May 2012, 05:10 PM Post #16 |
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Why's it derived from this? |
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A stranger has come To share my room in the house not right in the head. | |
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| Ardat | 3 May 2012, 05:12 PM Post #17 |
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Because what accurately models reality accurately models reality. I'm not sure what else you'd want to claim that a statement is reasonable. |
| What's more fun than fighting crime? | |
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| Categories+Sheaves | 3 May 2012, 05:46 PM Post #18 |
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Pedantic Curmudgeon
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Except that you need to have standards for comparing theory A's ability to model reality to theory B's ability to model reality. And those don't fall out of the sky either (in fact them tend to be packaged with our models of reality). And what are we referring to when we say "reality" anyway? I'm sure you would say vsaur's acid trip (where he meets god andmodern neuroscience/psychology is revealed to him as blasphemy) is less admissible than an EEG of his brain during said experience? My point is (not strictly fdrake's point, but it's probably analogous): if we're choosing between paradigm A and paradigm B, it isn't so much that one is right and the other is wrong, but more that we prefer interacting with the world in manner A to interacting with the world in manner B. As folks that like science, we should have pretty similar preferences in this regard, so I have a decent understanding of what you mean when you say "accurately model reality". But incommensurability is still a thing when we're talking about outsiders. |
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So these philosophers were all like, "That Kant apply universally!" And then these mathematicians were all like, "Oh yes it Kan!" Daily Dose of Zizek Invictus Fancypants Math Rants
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| Ardat | 4 May 2012, 05:41 AM Post #19 |
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Actually, I wouldn't say that. At least not if both are considered as mere observations. And my point is that the paradigm that's been shown to produce accurate predictions is the one that'll be perceived as reasonable. For an individual claim, its independent discoveries in paradigms that are not contingent upon each other makes it "reasonable" as well, if we acknowledge the unlikelihood of false converging statements in relatively unrelated framework. Sure, but I'm not necessarily limiting myself to the scientific method. To go back to a recurring theme, while refreshing fdrake's memory concerning his own thread, the reasonability of the claim "I have two feet" doesn't necessitate any kind of rigorous framework to be assumed, but merely a way for us to acknowledge our own engagement into the world, and a set of definitions derived from it. Edited by Ardat, 4 May 2012, 08:31 AM.
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| What's more fun than fighting crime? | |
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| fdrake | 4 May 2012, 10:34 AM Post #20 |
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Anyone can hide away in ambiguities when it suits them, I'm well practiced at it. ![]() Regardless, I didn't seem to convey my point adequately in the last thread. The background upon which we may say things like "I have two feet" is a set of ground rules for communication, these rules set the stage for communicating intelligibly. There's no "correct" way for the ground rules to go simpliciter, since that begs regress [who makes the rules?]. Nevertheless there are rules which arise - historically, conventionally, on occasion helped along with madcap ingenuity -, and ground rules like the modelling relation are definitely the standard for scientific discourse. What I'm seeking is a justification for the privilege of the modelling relation, you repeatedly keep saying "it works, look!", and that will convince no one who doesn't already believe in its relevence to the job at hand. You're attempting to change others' perceptual beliefs, here, by wild gesturing towards the way reality obviously is. To be sure, it's often an excellent thing to believe. I'd trust the physicist over the cobbler when talking about atoms, but the cobbler probably finds the crafting of shoes more obvious than the physicist. Essentially, ardat, given the above equivocation, you're setting ground rules for communication, rather than establishing their veracity - as if the rules themselves could be correct or incorrect, and what their regional ontologies* are are given solely by how these ground rules fit their styles of discourse. It seems like rationality only ever comes to bear when someone's wrong. ![]() You're playing a different game to most theists, then criticizing them by not playing by your rules... When you're doing precisely the same thing - refusing to look under the bonnet of the modelling relation, to examine it: twinned with many theists' failure to examine their own relation to the divine. To link this in with the Tortoise argument, these rules are part of the "magic" of communication. Any worldview wishing for consistency, presuming it values that and standard logical laws, ought be troubled by the Tortoise. We don't get to assume his questions out of the way. As you said, this equates to "shut up", otherwise known as not listening, when really all we say and do suggests we should be troubled by his questions. Sound familiar? regional ontology stuff
Edited by fdrake, 5 May 2012, 06:55 AM.
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