0816: "Applied Math"

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SocialSceneRepairman
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Re: 0816: "Applied Math"

Postby SocialSceneRepairman » Mon Nov 08, 2010 2:41 pm UTC

Ha, ha, guys. I meant that it's been proven that the set that would be made by such a proof breaks some fundamental principle of sets, so any proof that could show this wouldn't be recognizable as a valid proof as it's usually understood. I think Gödel himself proved that, actually? Should probably look this up. Not really my field.

reub
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Re: 0816: "Applied Math"

Postby reub » Mon Nov 08, 2010 3:04 pm UTC

"Beware of bugs in the above code; I have only proved it correct, not tried it" - Donald Knuth

mafaraxas
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Re: 0816: Applied Math

Postby mafaraxas » Mon Nov 08, 2010 3:23 pm UTC

DT_ wrote:
mafaraxas wrote:To be a small sticker on notation:

where it says [imath]P \Lambda \bar{P}[/imath], what you probably meant, the adjoint (conjugate transpose for finite-dimensional spaces), is usually written [imath]P^*[/imath] rather than [imath]\bar{P}[/imath] to distinguish it from only doing complex conjugate (without doing transpose) of all the elements in the matrix. So [imath]P \Lambda P^*[/imath] is how you usually see it. Unless this was the hidden joke and Randall was trying to troll mathematicians or something.

Also I don't really see what the comic has to do with applied math at all. Consistency of logic systems is about as far from applied math as you can get (restricting yourself to the realm of math). Unless this is part of the joke too? I don't know.


P is a proposition, not a matrix. The girl has proved "P and not P."


Well I guess you know what I've been seeing too much of :oops:

yhnmzw
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Re: 0816: Applied Math

Postby yhnmzw » Mon Nov 08, 2010 4:28 pm UTC

Waylah wrote:
Turing Machine wrote:http://plato.stanford.edu/entries/dialetheism/

seriously not terribly high-level logic here, guy


I was reading that link, and I got down to

(5) (5) is false, or neither true nor false, or the fourth thing.

And that's still supposed to be paradoxical, but I don't see how. Why can't 'the fourth thing' be a true thing? In which case, (5) is the fourth thing, and true. Which is not paradoxical.

Surely it would only be paradoxical if it was
(5) (5) is false, AND neither true nor false, or the fourth thing
or any other arrangement that says it can't be the fourth thing and true. Hahaha, what about
(5) (5) is false, or neither true nor false, or a paradox


To the contrary:
"The fourth thing" is meant to be another value in some unspecified logic system; "a paradox," for example, could be "the fourth thing."

Here, follow along with me:
The section is about trying to get away from paradoxes by using more possible "truth values" in a logic system. "True" and "false" alone failed to begin with, motivating the addition of "neither true nor false". This business of "the fourth thing" is supposed to cover things that don't want to be "true", "false", or "neither". Paradox (4) is such a thing.

Also, paradox can't be a truth value. You have to reach into the logic system and add "paradox" with your bare hands where they seem to pop up. If you do that, you can't say your logic is sound and reliable and academic.

Try counting from 1 to apple orchard. I don't know how you're going to move from numbers to concepts without a willful (mis)interpretation.

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Re: 0816: Applied Math

Postby Ghona » Mon Nov 08, 2010 6:16 pm UTC

yhnmzw wrote:Try counting from 1 to apple orchard. I don't know how you're going to move from numbers to concepts without a willful (mis)interpretation.

Do I have to explain everything to you nitwits? It's the simplest possible thing imaginable. First, convert from base 10 to base apple orchard. Second, start counting until you reach 10.
If you're taking me too seriously, you probably are making a mistake.

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Re: 0816: "Applied Math"

Postby neoliminal » Mon Nov 08, 2010 6:51 pm UTC

"It is better to be complete than consistent." -- John Lewis
http://www.amazon.com/dp/B0073YYXRC
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Re: 0816: "Applied Math"

Postby webgiant » Mon Nov 08, 2010 6:53 pm UTC

LucasBrown wrote:Image
Alt text: "Dear Reader: Enclosed is a check for ninety-eight cents. Using your work, I have proven that this equals the amount you requested."

The principle of explosion returns.

HEY! You forgot to carry the two!

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Tualha
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Re: 0816: "Applied Math"

Postby Tualha » Mon Nov 08, 2010 11:14 pm UTC

Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?

Nice of Knuth to send her $0.98. Could just as easily have said the number she claimed was equal to negative $10,000, so she owes him!

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Re: 0816: "Applied Math"

Postby xnick » Mon Nov 08, 2010 11:56 pm UTC

Tualha wrote:Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?


That'll just prove that your extended system of logic is inconsistent, not your starting system :(

The problem with the "usual" logic is that it is unknown whether is possible or not to derive a contradiction from the axioms
(and it is also impossible to know if such derivation will occur, and the problem extends to a general logic systems as well)

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Re: 0816: "Applied Math"

Postby Turing Machine » Tue Nov 09, 2010 12:13 am UTC

xnick wrote:
Tualha wrote:Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?


That'll just prove that your extended system of logic is inconsistent, not your starting system :(

The problem with the "usual" logic is that it is unknown whether is possible or not to derive a contradiction from the axioms
(and it is also impossible to know if such derivation will occur, and the problem extends to a general logic systems as well)


nope

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neoliminal
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Re: 0816: "Applied Math"

Postby neoliminal » Tue Nov 09, 2010 12:26 am UTC

fix'd
Image
http://www.amazon.com/dp/B0073YYXRC
Read My Book. Cost less than coffee. Will probably keep you awake longer.
[hint, scary!]

PokerJoker811
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Re: 0816: "Applied Math"

Postby PokerJoker811 » Tue Nov 09, 2010 2:44 am UTC

When you convert $3,372,564.48 into US dollars from the currency of the nation of San Serrife, then you do get 98 cents.

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Re: 0816: Applied Math

Postby NumberFourtyThree » Tue Nov 09, 2010 4:37 am UTC

Ezbez wrote:
SocialSceneRepairman wrote:Of course, hasn't it been proven that such a proof is impossible?


Assume such a proof exists. Then P and not P. Contradiction. Therefore, no such proof exists.

No, because if such a proof exists then contradictions are allowed and do not disprove things. You are begging the question by using reasoning based on logic being consistent to prove that logic can't be inconsistent.
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Tualha
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Re: 0816: "Applied Math"

Postby Tualha » Tue Nov 09, 2010 5:04 am UTC

xnick wrote:
Tualha wrote:Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?


That'll just prove that your extended system of logic is inconsistent, not your starting system :(


I just meant that in principle, you can prove at least some systems of logic to be inconsistent.

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Re: 0816: "Applied Math"

Postby Surg » Tue Nov 09, 2010 5:56 am UTC

Tualha wrote:
xnick wrote:
Tualha wrote:Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?


That'll just prove that your extended system of logic is inconsistent, not your starting system :(


I just meant that in principle, you can prove at least some systems of logic to be inconsistent.


This was known

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Re: 0816: "Applied Math"

Postby NumberFourtyThree » Tue Nov 09, 2010 6:36 am UTC

Tualha wrote:
xnick wrote:
Tualha wrote:Um, I haven't studied this much, but it seems obvious to me that you can prove a system of logic to be inconsistent. Take ordinary logic and add the axiom "P ^ ~P". Done. No?


That'll just prove that your extended system of logic is inconsistent, not your starting system :(


I just meant that in principle, you can prove at least some systems of logic to be inconsistent.

Only by using a different system of logic. If you formulate your proof in the same system of logic you have just proven your proof is invalid as it rests on an invalid system of logic.
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Re: 0816: "Applied Math"

Postby Kyrn » Tue Nov 09, 2010 8:15 am UTC

Instead of using the term invalid, just use the term inconsistent. Invalid can be mistaken to imply that it is always incorrect. Inconsistent is a more consistent term.

The whole point is proving that conventional logic is inconsistent by forming a paradox; a situation where it can neither be proven true or false. Conventional logic has no method of resolving such paradoxes, though extended logic may be able to.

I do not think it is possible to prove the non-existence of such logic.
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Re: 0816: "Applied Math"

Postby troyp » Tue Nov 09, 2010 12:47 pm UTC

<scoff> Everyone knows that although mathematics is inconsistent, any two contradictory statements are so computationally distant that their existence can only be confirmed using a special computer made of light...

Waylah
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Re: 0816: Applied Math

Postby Waylah » Tue Nov 09, 2010 12:54 pm UTC

yhnmzw wrote:
Waylah wrote:
Turing Machine wrote:http://plato.stanford.edu/entries/dialetheism/

seriously not terribly high-level logic here, guy


I was reading that link, and I got down to

(5) (5) is false, or neither true nor false, or the fourth thing.

And that's still supposed to be paradoxical, but I don't see how. Why can't 'the fourth thing' be a true thing? In which case, (5) is the fourth thing, and true. Which is not paradoxical.

Surely it would only be paradoxical if it was
(5) (5) is false, AND neither true nor false, or the fourth thing
or any other arrangement that says it can't be the fourth thing and true. Hahaha, what about
(5) (5) is false, or neither true nor false, or a paradox


To the contrary:
"The fourth thing" is meant to be another value in some unspecified logic system; "a paradox," for example, could be "the fourth thing."

Here, follow along with me:
The section is about trying to get away from paradoxes by using more possible "truth values" in a logic system. "True" and "false" alone failed to begin with, motivating the addition of "neither true nor false". This business of "the fourth thing" is supposed to cover things that don't want to be "true", "false", or "neither". Paradox (4) is such a thing.

Also, paradox can't be a truth value. You have to reach into the logic system and add "paradox" with your bare hands where they seem to pop up. If you do that, you can't say your logic is sound and reliable and academic.

Try counting from 1 to apple orchard. I don't know how you're going to move from numbers to concepts without a willful (mis)interpretation.


When I said "how about ... paradox" I did prefix it with 'haha'. But you totally missed my point; yes, as you say, " 'a paradox,' for example, could be 'the fourth thing.' " and that would work as a paradox, and yet not a paradox, but then yes it would have to be a paradox, but then that would mean it is not a paradox, etc. (hence the 'haha'). But 'the fourth thing' doesn't have to be 'a paradox'. There wasn't anything in point (5) that said the fourth thing couldn't allow the statement to be true also. Then the statement could be the fourth thing, and be true.
Granted, that doesn't get you very far, because all you need to do is change a carefully placed or to an and, and you're back to square one.

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Re: 0816: "Applied Math"

Postby Turing Machine » Tue Nov 09, 2010 1:15 pm UTC

Read the article next time.

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Tualha
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Re: 0816: "Applied Math"

Postby Tualha » Tue Nov 09, 2010 6:44 pm UTC

troyp wrote:<scoff> Everyone knows that although mathematics is inconsistent, any two contradictory statements are so computationally distant that their existence can only be confirmed using a special computer made of light...

Nice to see that someone else has read that story, but they really only needed that computer to map the entire defect. They found a starting point without it, remember?

For those who are wondering, the story is "Luminous" by Greg Egan. Also has a sequel, "Dark Integers".

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Re: 0816: "Applied Math"

Postby troyp » Tue Nov 09, 2010 8:46 pm UTC

Tualha wrote:Nice to see that someone else has read that story, but they really only needed that computer to map the entire defect. They found a starting point without it, remember?

Oh yeah, I remember now: they found the first anomaly with a years-long distributed computing effort.
Been a while since I last read Luminous.

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Re: 0816: "Applied Math"

Postby xnick » Tue Nov 09, 2010 10:20 pm UTC

Kyrn wrote: Inconsistent is a more consistent term.


LoL

Based on this I raise another question: are paradoxes ironic?

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Re: 0816: Applied Math

Postby Kyrn » Tue Nov 09, 2010 10:42 pm UTC

Waylah wrote:
yhnmzw wrote:
Waylah wrote:
Turing Machine wrote:http://plato.stanford.edu/entries/dialetheism/

seriously not terribly high-level logic here, guy


I was reading that link, and I got down to

(5) (5) is false, or neither true nor false, or the fourth thing.

And that's still supposed to be paradoxical, but I don't see how. Why can't 'the fourth thing' be a true thing? In which case, (5) is the fourth thing, and true. Which is not paradoxical.

Surely it would only be paradoxical if it was
(5) (5) is false, AND neither true nor false, or the fourth thing
or any other arrangement that says it can't be the fourth thing and true. Hahaha, what about
(5) (5) is false, or neither true nor false, or a paradox


To the contrary:
"The fourth thing" is meant to be another value in some unspecified logic system; "a paradox," for example, could be "the fourth thing."

Here, follow along with me:
The section is about trying to get away from paradoxes by using more possible "truth values" in a logic system. "True" and "false" alone failed to begin with, motivating the addition of "neither true nor false". This business of "the fourth thing" is supposed to cover things that don't want to be "true", "false", or "neither". Paradox (4) is such a thing.

Also, paradox can't be a truth value. You have to reach into the logic system and add "paradox" with your bare hands where they seem to pop up. If you do that, you can't say your logic is sound and reliable and academic.

Try counting from 1 to apple orchard. I don't know how you're going to move from numbers to concepts without a willful (mis)interpretation.


When I said "how about ... paradox" I did prefix it with 'haha'. But you totally missed my point; yes, as you say, " 'a paradox,' for example, could be 'the fourth thing.' " and that would work as a paradox, and yet not a paradox, but then yes it would have to be a paradox, but then that would mean it is not a paradox, etc. (hence the 'haha'). But 'the fourth thing' doesn't have to be 'a paradox'. There wasn't anything in point (5) that said the fourth thing couldn't allow the statement to be true also. Then the statement could be the fourth thing, and be true.
Granted, that doesn't get you very far, because all you need to do is change a carefully placed or to an and, and you're back to square one.

One significant problem with this set of logic: Analyze point 5 itself.
"(5) (5) is false, AND neither true nor false, or the fourth thing."
Note that it is phrased in an either-or format, which implies truth or non-truth. If we assume that there is such a value which represents neither true or false, (5) would STILL match that value (in a conventional logic sense, in a more accurate and extended sense, it neither matches or not-matches. See the pattern here?), because like (4), it is also neither true or false.

The problem is that we are explaining this extended logic using our common logic, which is inherently impossible without inconsistency, since the extended logic is developed solely to solve said inconsistency. Refer to that bracketed section.

EDIT: alternatively: consider this set: Inconsistent values. Why is it so hard to believe that this set can intersect itself?

Further consideration: Consider the Sets "True" and the Set "False", which comprises all possible logical statements. "Inconsistent" statements would fall under the overlap of "True" and "False".

Now consider the sets "Inconsistent: True", and "Inconsistent: False". Let us assume that there is a section of overlap between them where they are "Inconsistent". The issue is that this overlapped section would fall entirely under "Inconsistent: True" (Basically, there are no statements which are "Inconsistent: True" but not "Inconsistent: False"). Further analysis would note that "Inconsistent: False" is basically the combined sets of "True" and "False".

In short: All logical statements fall under sets "True" and "False", where set "Inconsistent" is defined as the overlapped members between "True" and "False".

LAST EDIT: And now, a thought exercise: There's actually a last section not considered here: Illogical statements such as "This sentence apple moo". This statement would technically fall under the section completely outside "True" and "False" (and hence outside "Inconsistent" as well). How does this section interact with the other sections?
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Zemyla
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Re: 0816: "Applied Math"

Postby Zemyla » Wed Nov 10, 2010 1:16 am UTC

Wait, she's cashing a Knuth reward check? No one cashes a Knuth reward check!

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Re: 0816: "Applied Math"

Postby HonoreDB » Wed Nov 10, 2010 1:38 am UTC

I'll have to check out those Greg Egan stories; I've really liked what I've read of him so far.

Ted Chiang has a story on the subject that I enjoyed, and it's online for free: Division By Zero.

Neoliminal's remix makes an interesting complement to the Chiang story.

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Re: 0816: "Applied Math"

Postby Turing Machine » Wed Nov 10, 2010 2:02 am UTC

HonoreDB wrote:I'll have to check out those Greg Egan stories; I've really liked what I've read of him so far.

Ted Chiang has a story on the subject that I enjoyed, and it's online for free: Division By Zero.

Neoliminal's remix makes an interesting complement to the Chiang story.


what's this guy from misetings doing here

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Re: 0816: "Applied Math"

Postby HonoreDB » Wed Nov 10, 2010 3:30 am UTC

Hey, no fair switching handles between fora.

distractedSofty
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Re: 0816: "Applied Math"

Postby distractedSofty » Wed Nov 10, 2010 6:29 am UTC

Zemyla wrote:Wait, she's cashing a Knuth reward check? No one cashes a Knuth reward check!


Well, since cashing it and not cashing it are equivalent...

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Re: 0816: Applied Math

Postby dindon » Wed Nov 10, 2010 6:37 am UTC

NumberFourtyThree wrote:
SocialSceneRepairman wrote:Of course, hasn't it been proven that such a proof is impossible?

You forget the subject. If logic itself is inconsistent, then an apparently solid proof that something can't be proven might be also provably false, as you could prove contradictory and false things. By the very nature of the question to prove that such a proof is impossible would be irrelevant as the production of such a proof would show that the proof of its inability to be proved was invalid.

A more troubling objection is that all proofs rely on basic logic, so such a proof of logic's invalidity would thus prove itself to be invalid and unable to be relied on to prove anything, including logic being invalid.

A logician draws a sharp distinction between the proof system he's dealing with and the metalanguage he's using to describe it (and to prove things about it). If I wanted to prove that the LK sequent calculus is sound and complete, I wouldn't write a proof using the sequent calculus. I would write a proof in English, using a variety of syntactic and semantic definitions which describe the proof system, but which are not a part of it.

When approached without rigor, logic is an area where it's easy to make spurious (but seemingly profound) statements.

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Re: 0816: "Applied Math"

Postby Ezbez » Wed Nov 10, 2010 12:53 pm UTC

I'm mildly disappointed that so many people took my "proof" on page 1 to be serious.

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Re: 0816: "Applied Math"

Postby xepher » Wed Nov 10, 2010 1:20 pm UTC

Hilbert cringes at Godel's Incompleteness Theorem.

This probably makes him roll in his grave.

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Re: 0816: "Applied Math"

Postby Poindexter » Wed Nov 10, 2010 6:23 pm UTC

What about T ^ ~T?

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Re: 0816: Applied Math

Postby Ghavrel » Wed Nov 10, 2010 9:28 pm UTC

dindon wrote:When approached without rigor, every subject is an area where it's easy to make spurious (but seemingly profound) statements.


I fixed that a bit for you. :wink:
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Re: 0816: Applied Math

Postby Turing Machine » Wed Nov 10, 2010 10:42 pm UTC

Ghavrel wrote:
dindon wrote:When approached without rigor, every subject is an area where it's easy to make spurious (but seemingly profound) statements.


I fixed that a bit for you. :wink:


lolsokalhoax

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Re: 0816: "Applied Math"

Postby rawrr » Thu Nov 11, 2010 7:45 pm UTC

not going to lie, i get really happy when randall makes all of the characters in a comic female.

i hate the view that "male is default", or even when there are a few (in the case of web comics) strips that do have some female characters there is at least one male present and, even funnier, there will be hundreds of other strips with no female characters. i didn't realize it until a little while ago that "all male = default, normal; all female = feminazis, crazy, wanting more than equality", but now i wonder how my female friends can stand reading/watching anything science/engineering oriented

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Re: 0816: "Applied Math"

Postby Optimystic » Thu Nov 11, 2010 7:56 pm UTC

Wow, girls suck at math.

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Re: 0816: Applied Math

Postby troyp » Thu Nov 11, 2010 8:00 pm UTC

dindon wrote:
NumberFourtyThree wrote:
SocialSceneRepairman wrote:Of course, hasn't it been proven that such a proof is impossible?

You forget the subject. If logic itself is inconsistent, then an apparently solid proof that something can't be proven might be also provably false, as you could prove contradictory and false things. By the very nature of the question to prove that such a proof is impossible would be irrelevant as the production of such a proof would show that the proof of its inability to be proved was invalid.

A more troubling objection is that all proofs rely on basic logic, so such a proof of logic's invalidity would thus prove itself to be invalid and unable to be relied on to prove anything, including logic being invalid.

A logician draws a sharp distinction between the proof system he's dealing with and the metalanguage he's using to describe it (and to prove things about it). If I wanted to prove that the LK sequent calculus is sound and complete, I wouldn't write a proof using the sequent calculus. I would write a proof in English, using a variety of syntactic and semantic definitions which describe the proof system, but which are not a part of it.

When approached without rigor, logic is an area where it's easy to make spurious (but seemingly profound) statements.

The premise of the comic is that the basic rules of logic that underlie all human reasoning don''t work. You can't just "step outside" of logic to use some metasystem to prove such a thing impossible.

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Re: 0816: Applied Math

Postby Turing Machine » Thu Nov 11, 2010 8:30 pm UTC

troyp wrote:
dindon wrote:
NumberFourtyThree wrote:
SocialSceneRepairman wrote:Of course, hasn't it been proven that such a proof is impossible?

You forget the subject. If logic itself is inconsistent, then an apparently solid proof that something can't be proven might be also provably false, as you could prove contradictory and false things. By the very nature of the question to prove that such a proof is impossible would be irrelevant as the production of such a proof would show that the proof of its inability to be proved was invalid.

A more troubling objection is that all proofs rely on basic logic, so such a proof of logic's invalidity would thus prove itself to be invalid and unable to be relied on to prove anything, including logic being invalid.

A logician draws a sharp distinction between the proof system he's dealing with and the metalanguage he's using to describe it (and to prove things about it). If I wanted to prove that the LK sequent calculus is sound and complete, I wouldn't write a proof using the sequent calculus. I would write a proof in English, using a variety of syntactic and semantic definitions which describe the proof system, but which are not a part of it.

When approached without rigor, logic is an area where it's easy to make spurious (but seemingly profound) statements.

The premise of the comic is that the basic rules of logic that underlie all human reasoning don''t work. You can't just "step outside" of logic to use some metasystem to prove such a thing impossible.


contradictions don't refute logic, hth

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Re: 0816: Applied Math

Postby troyp » Fri Nov 12, 2010 6:21 am UTC

Turing Machine wrote:contradictions don't refute logic, hth

The premise of the comic is certainly that it does: "You've shown the inconsistency - and thus invalidity - of basic logic itself".
This is true in that if you could prove P and not P, then you could no longer be confident that a valid chain of inference would constitute a sound argument.
From 'P and not P', you could infer any proposition at all, including a proof that you can't prove 'P and not P'. This is why NumberFourtyThree said such a proof would be irrelevant. dindon responded by talking about the distinction between a logic and a metalogic, presumably accusing NumberFourtyThree of assuming that the proof was formulated in the system under investigation.
In fact, there was no assumption about the proof. Any mathematical proof relies on basic logic, regardless of its formulation.
I can't believe I'm arguing about this


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