You lied to us (about universal truths in math)

For the discussion of math. Duh.

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Re: You lied to us (about universal truths in math)

Postby gmalivuk » Mon Jan 19, 2009 5:47 pm UTC

Viva El Shrooms wrote:but from what i can see the comic was still wrong, do you agree

The comic was a bloody comic. It wasn't meant to be well-reasoned philosophical commentary on the epistemological and ontological nature of mathematics...
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Re: You lied to us (about universal truths in math)

Postby skeptical scientist » Mon Jan 19, 2009 6:05 pm UTC

jestingrabbit wrote:everything has been reduced to set theory.
In some ways yes, in some ways, I'd say this is not at all true. The modern approach to mathematics is that yes, in principle, everything could be derived from the axioms of ZFC. But the general practice of doing mathematics seems to completely ignore this approach, at least from my perspective. We think of sets (and everything else for that matter) "naively", we use natural language arguments to prove theorems, and we generally assume that things could be done more formally, but it would be a bad idea.

Viva El Shrooms wrote:Ok I was wromg in my understanding of what I said, and thank you for explaining my misconceptions in mathmatics, but from what i can see the comic was still wrong, do you agree
No. I think math teachers do teach universal truths to their students. Of course, it is possible that not everything they teach is a universal truth, but I think that at least some of it is. Then again, I'm a mathematical Platonist; people with other philosophical viewpoints, including mathematicians, could disagree.
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Re: You lied to us (about universal truths in math)

Postby doogly » Mon Jan 19, 2009 6:07 pm UTC

Viva El Shrooms wrote:Ok I was wromg in my understanding of what I said, and thank you for explaining my misconceptions in mathmatics, but from what i can see the comic was still wrong, do you agree


No.
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Re: You lied to us (about universal truths in math)

Postby Yakk » Mon Jan 19, 2009 6:43 pm UTC

gmalivuk wrote:
Viva El Shrooms wrote:but from what i can see the comic was still wrong, do you agree

The comic was a bloody comic. It wasn't meant to be well-reasoned philosophical commentary on the epistemological and ontological nature of mathematics...

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Footnote [1]: The above sentence is intended to be a well-reasoned philosophical commentary on the epistemological and ontological nature of mathematics.

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Re: You lied to us (about universal truths in math)

Postby Certhas » Mon Jan 19, 2009 7:01 pm UTC

Viva El Shrooms wrote:Ok I was wromg in my understanding of what I said, and thank you for explaining my misconceptions in mathmatics, but from what i can see the comic was still wrong, do you agree



The universal truth taught is not: "a^2 + b^2 = c^2" but "Given the axioms of euclidean geometry, it follows that a^2 + b^2 = c^2".

The first statement is true, false or undecidable in different formal systems, the second statement is a universal truth. Or as the anglo saxon philosopher would have it: It's true in all possible worlds.
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Re: You lied to us (about universal truths in math)

Postby chapel » Mon Jan 19, 2009 9:25 pm UTC

I'll go with the roll over text. The joke is that if you are decent in something like political science or English, you can make a case for almost anything you want. For example, I wrote an essay in my 11th grade English final that explored the subtext of racism in the classic American literature "Green Eggs and Ham" by Dr. Suess. However, the distribution property of multiplication and addition can't really be politicized or refuted in any meaningful way by a high school student (unless they are exceptionally advanced and are familiar with RPN or Grassman numbers or some such).

Although, on facebook, there are two political groups surrounding the AC. So, do you feel that people should have the right to choose one element from an infinite collection of sets? Some people will have you believe that when no choice function exists that you should have your rights taken away from you.

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Re: You lied to us (about universal truths in math)

Postby diotimajsh » Mon Jan 19, 2009 10:01 pm UTC

Certhas wrote:The universal truth taught is not: "a^2 + b^2 = c^2" but "Given the axioms of euclidean geometry, it follows that a^2 + b^2 = c^2".

The first statement is true, false or undecidable in different formal systems, the second statement is a universal truth. Or as the anglo saxon philosopher would have it: It's true in all possible worlds.

This is very much my view. There may be philosophers and mathematicians who hold that a^2 + b^2 = c^2 simpliciter is a necessary truth (true in all possible worlds), but I think they haven't reflected enough on the matter. Similarly for a statement like 2+2=4: it's only true in the context of an appropriate system, but we're allowed to say that the proposition given certain assumptions (or a certain framework/system) universally follows.

We can then go on to speculate about the relations between mathematics/logic and the universe in a deeper way, about whether there's some metaphysical connection between 2+2=4 and the way that objects must behave in reality, or if there's a more "primitive" sense in which 2+2=4 is true, but I believe that's more the scope and interest of philosophy than mathematics. So we can still say that mathematicians teach universal truth, since they're not going around teaching metaphysical doctrines about number and reality*, but they are teaching techniques for objectively, universally evaluating statements within a well-defined framework.


* At least, not in an official capacity. As far as I know o.0
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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Mon Jan 19, 2009 10:17 pm UTC

Category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the field of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments.


That's interesting. Perhaps I'll look into category theory. I've heard, though, that it's so abstract as to be impenetrable; I wonder, can this be preferable to set theory?

Yakk wrote:Some people actually find those set theoretic definitions extremely illuminating and useful. There is an old joke in CS -- if you don't get CS until you run into the 5-tuple definitions of a FSM (states, starting, terminal, alphabet, transition function), then say "oh, so that is what it is", you might be suited to being a mathematician. Or words to that effect.


Ha! Perhaps I'm more suited to computer science than mathematics, then. I don't disagree that the 5-tuple definition makes great sense; it was just hard for me to keep in my head when I was first exposed to it. In retrospect, it's perfectly logical.

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Re: You lied to us (about universal truths in math)

Postby qinwamascot » Mon Jan 19, 2009 10:37 pm UTC

I don't see how the comic is wrong. It is a universal truth that, given axioms X, Y, and Z, we can draw some conclusions that will always hold. It depends on your definition of universal though. The truths are universal in that they apply to everything formulated in a certain way. But if you use the definition of universal that it is also unconditional, then it is not. I'd argue that a universal truth can still be conditional, but it's a semantic question and not entirely worth debating.
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Re: You lied to us (about universal truths in math)

Postby TheWaterBear » Mon Jan 19, 2009 10:42 pm UTC

IMO, just ask Randall what he really meant. :)

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Re: You lied to us (about universal truths in math)

Postby Certhas » Mon Jan 19, 2009 11:29 pm UTC

The Mad Scientist wrote:That's interesting. Perhaps I'll look into category theory. I've heard, though, that it's so abstract as to be impenetrable; I wonder, can this be preferable to set theory?


It depends on the question you are asking. Just as most mathematical questions are thought about in contexts far removed from those of set theory the same is true for category theory. That said, category theory can abstractly capture many mathematical properties that I as a theoretical physicist care about, so it feels like the right language when I want to go completely abstract. And the new perspectives allowed by this abstraction do prove usefull, which can't be said for chasing back to the abstract set theoretic underpinnings of these questions. Perhaps the most poignant example is Kindergarten QM. Some processes in QM really become simpler when looked at in this very abstract way.
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Re: You lied to us (about universal truths in math)

Postby doogly » Mon Jan 19, 2009 11:39 pm UTC

Certhas wrote:
The Mad Scientist wrote:That's interesting. Perhaps I'll look into category theory. I've heard, though, that it's so abstract as to be impenetrable; I wonder, can this be preferable to set theory?


It depends on the question you are asking. Just as most mathematical questions are thought about in contexts far removed from those of set theory the same is true for category theory. That said, category theory can abstractly capture many mathematical properties that I as a theoretical physicist care about, so it feels like the right language when I want to go completely abstract. And the new perspectives allowed by this abstraction do prove usefull, which can't be said for chasing back to the abstract set theoretic underpinnings of these questions. Perhaps the most poignant example is Kindergarten QM. Some processes in QM really become simpler when looked at in this very abstract way.


I agree, category theory is much more approachable than the set/foundation stuff. It is more like a step back, for extra perspective, than a step down, if this makes any sense.
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Re: You lied to us (about universal truths in math)

Postby t0rajir0u » Tue Jan 20, 2009 12:42 am UTC

The Mad Scientist wrote:Imagine taking an analysis course without ever having taken a course in calculus. ("Epsilon? Delta? What's all this about? Why should I care?") That's how most abstract mathematics is presented, unfortunately: all "analysis" and no "calculus".

Funnily enough, while I agree with your main point, I actually think this is how analysis should be taught - with the caveat that topology should be a prerequisite. Teaching calculus first provides intuition at the expense of the rigor that analysis deserves - when can you exchange limits? When can you differentiate a power series term-by-term? It's like teaching engineers how to build engines without teaching them about the second law of thermodynamics.

Ultimately, it depends on how seriously you want to take math - a mathematician should have a strong foundation in analysis and understand how the topology of the real line dictates real analysis, whereas an engineer doesn't need more than a good intuition about the fundamental theorem of calculus.

Perhaps to satisfy people who learn best differently, a really good integrated course should present both approaches simultaneously - historical development and modern axioms, plausible intuitions and rigorous proofs, lots of examples and theoretical exercises.

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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Tue Jan 20, 2009 1:10 am UTC

Funny that we can both be reasonable, t0rajir0u, and yet take stances diametrically opposed on the subject of how best to teach calculus and analysis. In my view, people learn better when you first show them the basic contours of an idea to give them a feel for it, then show them why the concept is interesting and useful, and only then introduce rigorous definitions. The danger of approaching instruction the opposite way is that many students will fail to see the forest for the trees, and conclude that the trees aren't so interesting!

However, I also believe that sequences and series should be given much more attention than they are currently, both at the upper level and at the lower level in the form of simple arithmetic and geometric sequences. I remember that as a child I loved the "guess the next number in the sequence" games that we sometimes played in math class, but then the whole subject was shelved until Calculus II! By then I had forgotten how fun and fascinating it was, and its reintroduction was a revelation. Infinite series became a unifying concept for me, one that helped me to better understand nearly every mathematical idea I'd ever been exposed to. (For example, the number e, which I once vaguely understood as the base of a strange exponential function which was its own derivative, became totally demystified and immediately made good sense.) Why do we neglect such a fundamental and interesting topic? Supposedly students have a hard time grasping series, but I've always found them to be intuitive and delightful. Am I in the minority?

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Re: You lied to us (about universal truths in math)

Postby auteur52 » Tue Jan 20, 2009 1:43 am UTC

The Mad Scientist wrote:Why do we neglect such a fundamental and interesting topic? Supposedly students have a hard time grasping series, but I've always found them to be intuitive and delightful. Am I in the minority?


I would say not. Rudin's chapter on Sequences and Series is possibly the longest in Baby Rudin, and also took up a majority of our first quarter of Analysis.

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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Tue Jan 20, 2009 2:13 am UTC

Allow me to clarify: I believe that sequences and series should be given more attention before one takes a course in analysis. By "lower levels" and "higher levels" I meant elementary algebra and calculus. I mentioned my opinion that series are neglected as an example of an exception to my opinion that rigorous analytical methods should be reserved until after an introductory calculus course. Sorry if I didn't explain myself correctly.

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Re: You lied to us (about universal truths in math)

Postby capefeather » Tue Jan 20, 2009 4:05 am UTC

I don't get what this guy is talking about. The comic says that math teachers can impart perfect universal truths to their students. There's not much more than that. Communication includes context as well as words. If someone says "all the dogs are brown" and gestures toward a room full of brown dogs, that's a universal truth (barring weird philosophical stuff). The conditions in which mathematicians are restricting themselves are always seen as implied through one means or another; otherwise, that's not a very good mathematical communicator.

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Re: You lied to us (about universal truths in math)

Postby Talith » Tue Jan 20, 2009 6:59 am UTC

capefeather wrote:I don't get what this guy is talking about. The comic says that math teachers can impart perfect universal truths to their students. There's not much more than that. Communication includes context as well as words. If someone says "all the dogs are brown" and gestures toward a room full of brown dogs, that's a universal truth (barring weird philosophical stuff). The conditions in which mathematicians are restricting themselves are always seen as implied through one means or another; otherwise, that's not a very good mathematical communicator.


I would argue that we can only know that the sides of the dogs which are in view are brown and that the other sides may be any other colour :) .

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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Tue Jan 20, 2009 7:05 am UTC

Talith wrote:
capefeather wrote:I don't get what this guy is talking about. The comic says that math teachers can impart perfect universal truths to their students. There's not much more than that. Communication includes context as well as words. If someone says "all the dogs are brown" and gestures toward a room full of brown dogs, that's a universal truth (barring weird philosophical stuff). The conditions in which mathematicians are restricting themselves are always seen as implied through one means or another; otherwise, that's not a very good mathematical communicator.


I would argue that we can only know that the sides of the dogs which are in view are brown and that the other sides may be any other colour :) .


You assume that they have other sides.

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Re: You lied to us (about universal truths in math)

Postby Talith » Tue Jan 20, 2009 7:31 am UTC

The Mad Scientist wrote:
Talith wrote:
capefeather wrote:I don't get what this guy is talking about. The comic says that math teachers can impart perfect universal truths to their students. There's not much more than that. Communication includes context as well as words. If someone says "all the dogs are brown" and gestures toward a room full of brown dogs, that's a universal truth (barring weird philosophical stuff). The conditions in which mathematicians are restricting themselves are always seen as implied through one means or another; otherwise, that's not a very good mathematical communicator.


I would argue that we can only know that the sides of the dogs which are in view are brown and that the other sides may be any other colour :) .


You assume that they have other sides.


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Re: You lied to us (about universal truths in math)

Postby Incompetent » Tue Jan 20, 2009 12:50 pm UTC

The Mad Scientist wrote:You assume that they have other sides.


+1 mathmo point for this.

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Re: You lied to us (about universal truths in math)

Postby itaibn » Wed Jan 21, 2009 1:58 am UTC

The Mad Scientist wrote:
Talith wrote:
capefeather wrote:I don't get what this guy is talking about. The comic says that math teachers can impart perfect universal truths to their students. There's not much more than that. Communication includes context as well as words. If someone says "all the dogs are brown" and gestures toward a room full of brown dogs, that's a universal truth (barring weird philosophical stuff). The conditions in which mathematicians are restricting themselves are always seen as implied through one means or another; otherwise, that's not a very good mathematical communicator.


I would argue that we can only know that the sides of the dogs which are in view are brown and that the other sides may be any other colour :) .


You assume that they have other sides.

Actually, I think he is trying to say that every other side may be any colour. If there isn't any other side, the statement is vacously (how do you spell that) true.
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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Wed Jan 21, 2009 4:57 am UTC

itaibn wrote:Actually, I think he is trying to say that every other side may be any colour. If there isn't any other side, the statement is vacously (how do you spell that) true.


Almost right, but wrong. You are trying to use the definition of p --> q, which is -p V q. This would be valid if he'd said, "If the dogs have other sides, they may be of any color." But he didn't say that; he said, "...the other sides may be any color."

Do you submit that "The present king of France is bald" is a vacuously true statement? I (and Bertrand Russel) would tend to disagree.

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Re: You lied to us (about universal truths in math)

Postby Certhas » Wed Jan 21, 2009 10:32 am UTC

However one could argue that "having another side" is analytically contained in "being a dog", in the same way as "has a lung and a brain" is contained in "bachelor" (at least until the robot revolution, so more specifically: "bachelor in 2009").

(Just saying that one could argue that, I can see counter arguments as well. Basically our problem is an unclear definition of "dog".)
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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Wed Jan 21, 2009 10:50 am UTC

No, because we don't see dogs, we see what appear to be dogs. By your logic, if I see a highly realistic cardboard cutout of a dog, I am justified in assuming that it has a tail, since "having a tail" is analytically contained in "being a dog". Never mind that it's not really a dog. ;)

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Re: You lied to us (about universal truths in math)

Postby Certhas » Wed Jan 21, 2009 12:00 pm UTC

That's the thing, it is not really a dog! Also you might erroneously call something that isn't a dog a dog because you got fooled, but once you saw that it was a cardboard cut out of a dog you would agree that it's not a dog because it doesn't have a backside.

So the question is, is it inherent in our conception of dog? In Kantian language that would mean having a backside is an analytic truth about language rather than an synthetic statement about reality, and can be considered true a priori.
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Re: You lied to us (about universal truths in math)

Postby SlyReaper » Wed Jan 21, 2009 12:02 pm UTC

The Mad Scientist wrote:No, because we don't see dogs, we see what appear to be dogs. By your logic, if I see a highly realistic cardboard cutout of a dog, I am justified in assuming that it has a tail, since "having a tail" is analytically contained in "being a dog". Never mind that it's not really a dog. ;)


Having a tail is not analytically contained in being a dog. Many dogs have their tails docked.

/pedantry
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Re: You lied to us (about universal truths in math)

Postby diotimajsh » Wed Jan 21, 2009 12:12 pm UTC

Certhas wrote:However one could argue that "having another side" is analytically contained in "being a dog", in the same way as "has a lung and a brain" is contained in "bachelor" (at least until the robot revolution, so more specifically: "bachelor in 2009").
Hmmmmm. I'm not convinced that "has a lung and brain" is analytically contained in the concept "bachelor." Couldn't we sensibly speak about there being bachelors in some radically different life-form that has no lungs at all? While this does of course turn into a semantics squabble, IMO being "male" and "unmarried" are the necessary and sufficient conditions for bachelorhood, but marriage and malehood need not be confined to humans or even mammals.

Well, a clarification. It might be better to revise "unmarried" to "unmarried but with the capacity to married," since otherwise we would be forced to say that every non-human male organism is a bachelor, since (I presume) the lot of them are not married. Also, we probably don't want to call children bachelors at least until they've reached the legal age of marriage, regardless of gender.

Admittedly, under ordinary circumstances we can safely assume that "having the capacity to be married" implies that the entity in question be human--it is a legal requirement, after all. But we do use marriage to describe fictitious non-human unions, as in many fantasy and science novels, the Redwall series, mythologies, etc., without it being a contradiction in terms; and it seems reasonable to me that, if we achieve successful strong AI in the real world, we might reclassify legal marriages to include all sentient beings, not just humans. (Eventually). This, to me, suggests that being human is not inherent in the notion of marriage or marriage requirements.

Lastly, it is debatable whether being human necessarily entails possessing a lung and brain at all. Presumably, we would not think that possessing an artificial lung rather than a biological one precludes one from humanity, would we? And perhaps the brain of an ordinary human could be replaced by a fully functioning cybernetic equivalent without disqualifying him/her from humanity either?

...Which is all to say, we cannot deduce from thinking about the concept "bachelor" alone that the bachelor in question possesses lungs and a brain. We can infer that it's probably the case, based on other facts that we know, but it is not a necessary or analytic consequence of the concept bachelor.
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Re: You lied to us (about universal truths in math)

Postby chapel » Wed Jan 21, 2009 12:40 pm UTC

Due to being kept in an oddly shaped cage sense he was a puppy by a past cruel owner, my dog's spine is a non-orientable loop with Euler characteristic 0. I named him Moby.

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Re: You lied to us (about universal truths in math)

Postby Certhas » Wed Jan 21, 2009 2:05 pm UTC

SlyReaper wrote:
The Mad Scientist wrote:No, because we don't see dogs, we see what appear to be dogs. By your logic, if I see a highly realistic cardboard cutout of a dog, I am justified in assuming that it has a tail, since "having a tail" is analytically contained in "being a dog". Never mind that it's not really a dog. ;)


Having a tail is not analytically contained in being a dog. Many dogs have their tails docked.

/pedantry


That's why I used "has a backside".

Anzwaz, all the analytic/synthetic distinctions are somewhat pedantic and arbitrary, more rigorous than natural language certainly. Would you call a dog with amputated hind legs a dog? With amputated behind? with body replaced by a robot? with it's brain transferred into a machine? What about a bachelor?

What if we have a species with three genders, and the gender roles matching up in no clear way to out 2 gender system? Could one still speak of bachelors without a concept of male then?

As to the lungs and brains issue, that's why I specified in 2008 because at the moment the technology does not exist, but that also shows I guess that it is synthetic in the end...
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Re: You lied to us (about universal truths in math)

Postby itaibn » Wed Jan 21, 2009 9:13 pm UTC

The Mad Scientist wrote:
itaibn wrote:Actually, I think he is trying to say that every other side may be any colour. If there isn't any other side, the statement is vacously (how do you spell that) true.


Almost right, but wrong. You are trying to use the definition of p --> q, which is -p V q. This would be valid if he'd said, "If the dogs have other sides, they may be of any color." But he didn't say that; he said, "...the other sides may be any color."

Notice that he said "the other sides". That implies that he meant this to be true for every other side. Thus, my argument holds.
edit/add: He was pretty much saying "for every object, if it is another side of one of the dogs, it may be any other colour". It might be vacicously true.
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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Wed Jan 21, 2009 9:36 pm UTC

SlyReaper wrote:Having a tail is not analytically contained in being a dog. Many dogs have their tails docked.

/pedantry


You are of course correct. The point is that until we know that we're looking at a dog, we can't rightly speak of any of its characteristics which are not immediately visible, be they analytically contained in the concept of "dog" or not.

itaibn wrote:Notice that he said "the other sides". That implies that he meant this to be true for every other side. Thus, my argument holds.
edit/add: He was pretty much saying "for every object, if it is another side of one of the dogs, it may be any other colour". It might be vacicously true.


No, what he was saying was the conjunction of the following two propositions:

1. For every visible dog, that dog has at least one side other than the one which is visible.
2. If an object is one of those other sides, it may be any other color.

The first proposition is an assumption. The point is that when you say "X may be Y", you are really saying, "X exists, and may be Y."

Without this distinction we would be forced to give truth values to statements with no referents. I refuse to believe that "The present king of France is bald" is true, vacuously or not, because there is no present king of France! Things do not change if you say, "The present king of France is male", even though being male is analytically contained in the concept of being a king. The only way things change is if you say, "If there were a king of France, he would be male." Now, you would have me believe that a statement such as, "The king of every country is male" is vacuously true, since you think that it takes the form of (only) proposition 2 above. What I'm trying to explain is that such a statement assumes that every country has a king.

Just for fun, let me throw in a blatant appeal to emotion. How would you feel if, in open court, a lawyer declared, in the course of cross-examining you, "SlyReaper, everyone knows that every woman you raped was female."

:P

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Re: You lied to us (about universal truths in math)

Postby capefeather » Wed Jan 21, 2009 11:40 pm UTC

Wow. This thread sure changed topics pretty quickly.

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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Thu Jan 22, 2009 12:16 am UTC

For this I feel no remorse, since the thread's original subject has already been settled. The original poster has recanted his thesis.

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Re: You lied to us (about universal truths in math)

Postby capefeather » Thu Jan 22, 2009 12:54 am UTC

I didn't say it was bad :D

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Re: You lied to us (about universal truths in math)

Postby diotimajsh » Thu Jan 22, 2009 12:58 am UTC

Certhas wrote:Anzwaz, all the analytic/synthetic distinctions are somewhat pedantic and arbitrary, more rigorous than natural language certainly. Would you call a dog with amputated hind legs a dog? With amputated behind? with body replaced by a robot? with it's brain transferred into a machine? What about a bachelor?
I agree that's it's pedantic, and these kind of questions pursued too far lead us to the notion that everyday words break if you push them too far, a la Paul Graham. (I don't, incidentally, agree with many other parts of that essay, but I do agree with that aspect.)

In any case, I'll second your earlier conclusion that "Basically our problem is an unclear definition of 'dog'". And that should teach us a lesson for trying to inappropriately use a natural language concept in a mathematical context. =P

Certhas wrote:As to the lungs and brains issue, that's why I specified in 2008 because at the moment the technology does not exist, but that also shows I guess that it is synthetic in the end...
Yeah, I dunno. As far as I can tell, the only way to make a truly analytic statement is to specify it so completely that it's an overt tautology. E.g., it will wind up being "A bachelor with lungs and brains has lungs and brains" or equivalent, though perhaps less explicitly.

...And of course, there's Quine's notorious critique of the analytic/synthetic distinction itself, but honestly I don't understand it very well...
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Re: You lied to us (about universal truths in math)

Postby Certhas » Thu Jan 22, 2009 2:50 pm UTC

The Mad Scientist wrote:Just for fun, let me throw in a blatant appeal to emotion. How would you feel if, in open court, a lawyer declared, in the course of cross-examining you, "SlyReaper, everyone knows that every woman you raped was female."

:P


But that is true for two reasons: Women are female, but if he didn't rape anyone it would still be true if you said "every woman you raped was blond" since you are talking about no one!

"Every person considered by the majority of French to be the King of France is wearing pink slippers", is vacuously true, not for being tautological or analytic but for referring to no object whatsoever. Abstractly: "Every element of the empty set has property X" is vacuously true.

"Every element of A has property X" could be translated as: "Take an object. If the object is in A it has property X. If it is not in A we have said nothing about it." If A is the empty set it thus translates to: "Take an object. We have said nothing about it."
"I conclude that all is well," says Edipus, and that remark is sacred.
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Re: You lied to us (about universal truths in math)

Postby Yakk » Thu Jan 22, 2009 5:50 pm UTC


As a mathematician, that is easy to solve. I define "dog" to be anything that barks at cats, chases cars, is perfectly spherical and a perfect black-body radiator. This is distinct from "cow"s, which do not bark at cats and may or may not chase cars.
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Re: You lied to us (about universal truths in math)

Postby diotimajsh » Thu Jan 22, 2009 8:49 pm UTC

Yakk wrote:As a mathematician, that is easy to solve. I define "dog" to be anything that barks at cats, chases cars, is perfectly spherical and a perfect black-body radiator. This is distinct from "cow"s, which do not bark at cats and may or may not chase cars.

Sea kittens!*


* The above statement is intended as a well-reasoned, in-depth rejection of your definition on the grounds that it grievously offends both common decency and intellectual aesthetics. :P

Certhas wrote:"Every element of A has property X" could be translated as: "Take an object. If the object is in A it has property X. If it is not in A we have said nothing about it." If A is the empty set it thus translates to: "Take an object. We have said nothing about it."
And of course then, this allows us to make vacuously true but seemingly paradoxical statements, neh?

"Every woman currently living on Pluto is male."

"Every true falsity is false."
"Every false truth is true."

"Every circular square is neither circular nor square." (This one almost makes an intuitive sense... but then, it really doesn't, when we consider that every circular square is also circular and square.)

"Every present King of France is simultaneously bald and not bald."
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Re: You lied to us (about universal truths in math)

Postby The Mad Scientist » Thu Jan 22, 2009 10:00 pm UTC

Certhas wrote:
The Mad Scientist wrote:Just for fun, let me throw in a blatant appeal to emotion. How would you feel if, in open court, a lawyer declared, in the course of cross-examining you, "SlyReaper, everyone knows that every woman you raped was female."

:P


But that is true for two reasons: Women are female, but if he didn't rape anyone it would still be true if you said "every woman you raped was blond" since you are talking about no one!

"Every person considered by the majority of French to be the King of France is wearing pink slippers", is vacuously true, not for being tautological or analytic but for referring to no object whatsoever. Abstractly: "Every element of the empty set has property X" is vacuously true.

"Every element of A has property X" could be translated as: "Take an object. If the object is in A it has property X. If it is not in A we have said nothing about it." If A is the empty set it thus translates to: "Take an object. We have said nothing about it."


Vacuous truth is a convenient fiction used to obtain a stronger logical system than those which, strictly speaking, make more sense. One must only admit its usefulness, much like one must only admit the usefulness of saying that .99999... = 1.

I myself am a fan of relevance logic, but it's too weak to be useful. Still, I will never admit that a "vacuous" truth is a real truth.


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