0263: "Certainty"

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Shpadoinkle
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Postby Shpadoinkle » Sun May 20, 2007 6:33 pm UTC

wibblymat wrote:The alt text needs prefixing with "For all a,b,c in a field," or similar. Maths is only as perfect as the way you write it down!


No, math remains perfect. The problem lies in human error and/or the inability to properly understand it.

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Postby JoshuaZ » Sun May 20, 2007 6:40 pm UTC

Stig Hemmer wrote:
skeptical scientist wrote:No, I'm referring to complete extensions of PA (assuming it's consistent) - i.e. you take the theory of PA, and add consistent statements to it until it is complete. Godel's theorem tells you this can't be done by adding a computable or even computably enumerable set of axioms, but it is theoretically possible. You just list all of the sentences of the language of arithmetic, and add each, or its negation, to your theory; at least one of the two must be consistent, so this can be done until every sentence or its negation is in your theory.


The problem is that you have to choose between adding a sentence or its negation.

If you specify a method to make this choice, you are vulnerable to Gödels attack and your theory has to inconsistent (since it can't be incomplete)

On the other hand, if you don't specify a method to make this choice, you don't really have a theory.

Another reason to reject the Axiom of Choice.:roll:


No, you have a theory, but the statements of the theory are not recursively enumerable. It makes it a boring theory but still a theory.

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Postby skeptical scientist » Sun May 20, 2007 8:51 pm UTC

Also, for what it's worth, you don't need the axiom of choice to construct this theory. You just take the sentence if consistent, or the negation if the sentence would be inconsistent. To carry out the construction you would need some way of determining whether a general sentence was consistent, which would probably require an oracle for the halting problem or something other way of gaining extra information, but to actually define the construction you don't need any special tools, even the axiom of choice. So such a theory does exist and is perfectly good for theoretic purposes - it's interesting, for example, to talk about properties of complete extensions of PA.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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Cyberax
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Postby Cyberax » Mon May 21, 2007 2:10 am UTC

skeptical scientist wrote:You just list all of the sentences of the language of arithmetic, and add each, or its negation, to your theory; at least one of the two must be consistent, so this can be done until every sentence or its negation is in your theory.

This can't be done in finite time - there are infinitely many combinations of words from theory's alphabet.

The next problem is the method of listing of sentences. Basically, you can do it in two ways:

1. Take a valid expression (like 1+1=2) and start to 'mutate' it: 1+1-1=2-1, 1=2-1*1, .... This process eventually enumerates all possible proofs in PA. But Goedel's theorem also states that there are true expressions in PA which can't be proved by a finite number of applications of rules of inference in this theory. So this algorithm won't enumerate all expressions in PA.

2. Just try to list all possible alphabet combinations in PA. This algorithm will eventually enumerate all possible combinations in PA. But there's another small problem - how would you decide which expressions you should include in the axiomatic set? Generally, you can't determine if a statement is contradictory.

Actually, I had this very discussion with my friend some time ago and I proposed this very method of creating a complete theory :) He won that argument.

BTW, sorry for my English (especially math terms) - it's not my native language.

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Postby Ukridge » Mon May 21, 2007 2:19 am UTC

Jwink3101 wrote:With saying it isn't a science, i do acknowledge that it is the language of all physics and therefore the language of all science (biology is applied chemistry, chemistry is applied physics, physics is applied math). That is to say, physics is applied math like literature is applied language.


"psychology is not applied biology nor is biology applied chemistry
whats the issue?
psychology is not applied biology nor is biology applied chemistry
whats the issue?
psychology is not applied biology nor is biology applied chemistry
whats the issue?
psychology is not applied biology nor is biology applied chemistry
whats the issue?
psychology is not applied biology nor is biology applied chemistry
whats the issue?"
-Doctor Octagon

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Postby JoshuaZ » Mon May 21, 2007 4:58 am UTC

Cyberax wrote:
skeptical scientist wrote:You just list all of the sentences of the language of arithmetic, and add each, or its negation, to your theory; at least one of the two must be consistent, so this can be done until every sentence or its negation is in your theory.

This can't be done in finite time - there are infinitely many combinations of words from theory's alphabet.


Irrelevant, this is a theoretical construction, not a construction take were doing in real life. Unless one has some sort of extreme intuitionistic attitude this doesn't matter.

The next problem is the method of listing of sentences. Basically, you can do it in two ways:

Cyberax wrote:1. Take a valid expression (like 1+1=2) and start to 'mutate' it: 1+1-1=2-1, 1=2-1*1, .... This process eventually enumerates all possible proofs in PA. But Goedel's theorem also states that there are true expressions in PA which can't be proved by a finite number of applications of rules of inference in this theory. So this algorithm won't enumerate all expressions in PA.

2. Just try to list all possible alphabet combinations in PA. This algorithm will eventually enumerate all possible combinations in PA. But there's another small problem - how would you decide which expressions you should include in the axiomatic set? Generally, you can't determine if a statement is contradictory.


Well, really one would do the second, since one can tell what is a WFF(that is a meaningful sequence of symbols. "1=2" is meaningful "===+" is not) is so one would just go through some lexicographic ordering of the WFFs, but you seem to be missing the point. This isn't intended as a construction that a human can do, but as a construction we can specify. That's the relevant issue. To make this slightly more explicit,
consider the following procedure which we can't do recursively but can specify.

Set T_0= PA with nice alphabet Sigma
Let n_i for i>0 be some enumeration of all WFFs in our formulation of PA.
Now, at stage i, define T_i = T_(i-1) if n_i or ~n_i is provable in T_i, otherwise add n_i as an axiom (we could just as easily flip a coin, or add ~n_i but this is the easiest way of doing things).
Let T_oo be the system that has every axiom that was an axiom of any T_i. T_00 is a complete, consistent extension of PA.

Here's the important part: The provable statements of T_i are not recursively enumerable (since no recursive procedure can tell us in general whether or not a given statement n_i is provable in T_i). So Godel's theorem does not apply.

Cyberax wrote:
Actually, I had this very discussion with my friend some time ago and I proposed this very method of creating a complete theory :) He won that argument.



Too bad. He was wrong.

Cyberax wrote:BTW, sorry for my English (especially math terms) - it's not my native language.


English seemed fine.

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Postby Cyberax » Mon May 21, 2007 11:58 am UTC

JoshuaZ wrote:This can't be done in finite time - there are infinitely many combinations of words from theory's alphabet.


Irrelevant, this is a theoretical construction, not a construction take were doing in real life. Unless one has some sort of extreme intuitionistic attitude this doesn't matter.
[/quote]
Personally, I prefer my axiomatic systems to be at least recursively enumerated :)

Well, really one would do the second, since one can tell what is a WFF(that is a meaningful sequence of symbols. "1=2" is meaningful "===+" is not) is so one would just go through some lexicographic ordering of the WFFs, but you seem to be missing the point. This isn't intended as a construction that a human can do, but as a construction we can specify. That's the relevant issue. To make this slightly more explicit,
consider the following procedure which we can't do recursively but can specify.

I'm not speaking about syntactic correctness - it's easy to check (or even to create a generator which outputs only valid expressions).

Now, at stage i, define T_i = T_(i-1) if n_i or ~n_i is provable in T_i, otherwise add n_i as an axiom

Ok, got it.

Still, it requires (in general) an infinite number of steps to check if n_i is provable.

The provable statements of T_i are not recursively enumerable (since no recursive procedure can tell us in general whether or not a given statement n_i is provable in T_i). So Godel's theorem does not apply.

I'm curious, what can you do with such axiomatic system?

Cyberax wrote:Actually, I had this very discussion with my friend some time ago and I proposed this very method of creating a complete theory :) He won that argument.

Too bad. He was wrong.

We argued only about nice recursively enumerated theories :)

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Postby skeptical scientist » Mon May 21, 2007 2:28 pm UTC

He's right - there is no recursively enumerable complete extension of PA. There is however a complete extension which is not r.e. One can prove their existence by giving a construction like the one I gave informally and JoshuaZ gave formally. The other question is whether you can actually find one effectively. The answer to this is obviously no, unless you have an oracle for the halting problem or some other method of hypercomputation. It's sort of like the halting problem - there is a set which consists of all of the ordered pairs (Turing machine, input) so that the specified machine does not halt on the specified input, but you can't actually build such a set.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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Postby Princess_Shauna » Mon May 21, 2007 3:54 pm UTC

So I could Definitly see a few of my teachers doing this.

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Postby svk » Mon May 21, 2007 8:07 pm UTC

I concur that the alt text should be made into a t-shirt. The only problem is that I intend to wear this to school. Perchance there could be a "sanitized" version of the shirt for people in a predicament such as mine?

Oh, and make it snappy. I only have about 15 days before I never have to see my government teacher again.

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Postby skeptical scientist » Mon May 21, 2007 8:54 pm UTC

There's nothing stopping you from going to a custom t-shirt place and asking them to make you a t-shirt with that as a slogan.

Well, nothing but laziness, anyways.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

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Postby JoshuaZ » Tue May 22, 2007 12:44 am UTC

svk wrote:I concur that the alt text should be made into a t-shirt. The only problem is that I intend to wear this to school. Perchance there could be a "sanitized" version of the shirt for people in a predicament such as mine?

Oh, and make it snappy. I only have about 15 days before I never have to see my government teacher again.


Are you in an American public school? You might be able to argue that the shirt fell within protected free speech since it is making a political argument (i.e. that certain things aren't political). Arguably that would politicize the statement itself and defeat the purpose...

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Postby saibot834 » Wed May 23, 2007 5:04 pm UTC

This comic is very funny :)

Today was my math test (I'm still in school) and I printed out the comic and gave it back along with my test... let's see if my math teacher thinks too, that it's funny and perhaps he will give me some extra points :)

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Postby bbctol » Wed May 23, 2007 7:52 pm UTC

saibot834 wrote:This comic is very funny :)

Today was my math test (I'm still in school) and I printed out the comic and gave it back along with my test... let's see if my math teacher thinks too, that it's funny and perhaps he will give me some extra points :)


Is there a reason for the word "bot" in your name? Or should I be nervous?

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Postby Yakk » Wed May 23, 2007 7:53 pm UTC

JoshuaZ wrote:
Cyberax wrote:
skeptical scientist wrote:You just list all of the sentences of the language of arithmetic, and add each, or its negation, to your theory; at least one of the two must be consistent, so this can be done until every sentence or its negation is in your theory.

This can't be done in finite time - there are infinitely many combinations of words from theory's alphabet.


Irrelevant, this is a theoretical construction, not a construction take were doing in real life. Unless one has some sort of extreme intuitionistic attitude this doesn't matter.

The next problem is the method of listing of sentences. Basically, you can do it in two ways:

Cyberax wrote:1. Take a valid expression (like 1+1=2) and start to 'mutate' it: 1+1-1=2-1, 1=2-1*1, .... This process eventually enumerates all possible proofs in PA. But Goedel's theorem also states that there are true expressions in PA which can't be proved by a finite number of applications of rules of inference in this theory. So this algorithm won't enumerate all expressions in PA.

2. Just try to list all possible alphabet combinations in PA. This algorithm will eventually enumerate all possible combinations in PA. But there's another small problem - how would you decide which expressions you should include in the axiomatic set? Generally, you can't determine if a statement is contradictory.


Well, really one would do the second, since one can tell what is a WFF(that is a meaningful sequence of symbols. "1=2" is meaningful "===+" is not) is so one would just go through some lexicographic ordering of the WFFs, but you seem to be missing the point. This isn't intended as a construction that a human can do, but as a construction we can specify. That's the relevant issue. To make this slightly more explicit,
consider the following procedure which we can't do recursively but can specify.

Set T_0= PA with nice alphabet Sigma
Let n_i for i>0 be some enumeration of all WFFs in our formulation of PA.
Now, at stage i, define T_i = T_(i-1) if n_i or ~n_i is provable in T_i, otherwise add n_i as an axiom (we could just as easily flip a coin, or add ~n_i but this is the easiest way of doing things).
Let T_oo be the system that has every axiom that was an axiom of any T_i. T_00 is a complete, consistent extension of PA.

Here's the important part: The provable statements of T_i are not recursively enumerable (since no recursive procedure can tell us in general whether or not a given statement n_i is provable in T_i). So Godel's theorem does not apply.


Demonstrate something concrete that would be true if your system is consistent and false if not?

You did a phase-shift from a system with a finite set of axioms to one with an infinite set of axioms. Meta-axioms about finite axiom systems and what consistency means seem reasonable -- meta-axioms about non-constructable infinite axiom systems and what it means for them to be consistent seem useless to me, and I wouldn't be surprised if you could consistently define your systems to be inconsistent without making sane-axiom logic theory go bonkers.

Basically, I suspect you have wandered off into the realm of "take a sphere, cut it up, rotate, translate, get 2 spheres", in which the truth of your theorum depends on axioms which you can flip-flop either way.

I don't know about you, but I don't know what it means to have a logic system in which the axioms are provably non-computable to be "complete".

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Postby JoshuaZ » Wed May 23, 2007 8:51 pm UTC

Yakk wrote:
JoshuaZ wrote:
Cyberax wrote:
skeptical scientist wrote:You just list all of the sentences of the language of arithmetic, and add each, or its negation, to your theory; at least one of the two must be consistent, so this can be done until every sentence or its negation is in your theory.

This can't be done in finite time - there are infinitely many combinations of words from theory's alphabet.


Irrelevant, this is a theoretical construction, not a construction take were doing in real life. Unless one has some sort of extreme intuitionistic attitude this doesn't matter.

The next problem is the method of listing of sentences. Basically, you can do it in two ways:

Cyberax wrote:1. Take a valid expression (like 1+1=2) and start to 'mutate' it: 1+1-1=2-1, 1=2-1*1, .... This process eventually enumerates all possible proofs in PA. But Goedel's theorem also states that there are true expressions in PA which can't be proved by a finite number of applications of rules of inference in this theory. So this algorithm won't enumerate all expressions in PA.

2. Just try to list all possible alphabet combinations in PA. This algorithm will eventually enumerate all possible combinations in PA. But there's another small problem - how would you decide which expressions you should include in the axiomatic set? Generally, you can't determine if a statement is contradictory.


Well, really one would do the second, since one can tell what is a WFF(that is a meaningful sequence of symbols. "1=2" is meaningful "===+" is not) is so one would just go through some lexicographic ordering of the WFFs, but you seem to be missing the point. This isn't intended as a construction that a human can do, but as a construction we can specify. That's the relevant issue. To make this slightly more explicit,
consider the following procedure which we can't do recursively but can specify.

Set T_0= PA with nice alphabet Sigma
Let n_i for i>0 be some enumeration of all WFFs in our formulation of PA.
Now, at stage i, define T_i = T_(i-1) if n_i or ~n_i is provable in T_i, otherwise add n_i as an axiom (we could just as easily flip a coin, or add ~n_i but this is the easiest way of doing things).
Let T_oo be the system that has every axiom that was an axiom of any T_i. T_00 is a complete, consistent extension of PA.

Here's the important part: The provable statements of T_i are not recursively enumerable (since no recursive procedure can tell us in general whether or not a given statement n_i is provable in T_i). So Godel's theorem does not apply.


Demonstrate something concrete that would be true if your system is consistent and false if not?



I'm not sure what you mean by this. The system is consistent.

Yakk wrote:You did a phase-shift from a system with a finite set of axioms to one with an infinite set of axioms.


Really? People are fine with systems that have axiom schema all the time. Even ZF has them.

Yakk wrote:Meta-axioms about finite axiom systems and what consistency means seem reasonable -- meta-axioms about non-constructable infinite axiom systems and what it means for them to be consistent seem useless to me, and I wouldn't be surprised if you could consistently define your systems to be inconsistent without making sane-axiom logic theory go bonkers.

Basically, I suspect you have wandered off into the realm of "take a sphere, cut it up, rotate, translate, get 2 spheres", in which the truth of your theorum depends on axioms which you can flip-flop either way.

I don't know about you, but I don't know what it means to have a logic system in which the axioms are provably non-computable to be "complete".


The system is consistent because one cannot derive A^~A in the system for any A. The system is complete because for any WFF A, either A is a theorem of the system or ~A is a theorem. That's the defintion of completeness. The term your looking for is sometimes "axiomatizable" which means that all valid statements in the theory are recursively enumerable. Complete and axiomatizable are different issues. One might not like systems that are complete and non-axiomatizable but they exist and we can prove metamathematical statements about them just as we can with axiomatizable theories. Consider the following analogy, there are transcendental real numbers whose digits are not recursively enumerable. However, we can prove results about those numbers such as a variety of results in the field of Diophantine approximation. These axiomatic systems are similar, we can prove results about them such as non-trivial things about minimal proof length of statements even though we cannot recursively describe the systems.

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Postby skeptical scientist » Wed May 23, 2007 8:52 pm UTC

What do you mean, "demonstrate one thing that is true if your system is consistent and false if not?" It is hard to describe two more different systems - in one, it is possible to prove any logical statement, and in the other, you can only prove a statement or its negation, and not both. Since systems are principally concerned with what you can prove and what you can't, this would seem to be the ballgame.

I'll assume you meant to ask why we want to look at complete extensions of theories. You are right - on the face of it, such theories would not seem terribly useful as they can't be used to prove things, since you never know what your axioms are in any real sense.

One extremely important theorem in model theory is the completeness theorem, (one version of) which states that every consistent theory has a model. The existence of models is a fundamental result and it would be very difficult to prove any of the standard independence results (such as the independence of the axiom of choice and the continuum hypothesis) without it. But it is far easier to prove that it is true by taking complete extensions (the standard proof - although not the original - is by adding in a bunch of constant symbols and what are called Henkin axioms, taking a completion of the resulting theory, and using this to define a model). This is actually a standard way completions of theories are used - they aren't usually studied as intrinsic objects in themselves, unless you happen to have a nice theory which is already complete, but are rather used as tools to deduce some useful result about arbitrary theories, which may or may not be complete. Complete theories are easier to work with for some purposes, but since every consistent theory has a complete extension, you can prove results about arbitrary theories using results about complete theories. So they are useful, even if an arbitrary badly non-computable theory is itself not something you can work in.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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Postby nosh276 » Thu May 24, 2007 5:05 pm UTC

mikesty wrote:
And, uh, all of that debate stuff, and the concrete beauty of math.


Math only seems concrete. Numbers are just as arbitrary as words. They have no definite meaning. Reference Godel's incompleteness theorem.

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Postby JoshuaZ » Thu May 24, 2007 6:55 pm UTC

nosh276 wrote:
mikesty wrote:
And, uh, all of that debate stuff, and the concrete beauty of math.


Math only seems concrete. Numbers are just as arbitrary as words. They have no definite meaning. Reference Godel's incompleteness theorem.


: Hmm, someone hasn't been following the thread....

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Postby Cyberax » Thu May 24, 2007 7:19 pm UTC

Demonstrate something concrete that would be true if your system is consistent and false if not?

Well, this system is complete by construction. So you can prove (or disprove) any statement within this system (because all unprovable statements are used as axioms).

You did a phase-shift from a system with a finite set of axioms to one with an infinite set of axioms.

Infinite sets are OK, Peano Arithmetics also uses them.

The difference between the constructed complete system and PA is that I can write schema of PA on a sheet of paper (it consists of a finite number of recursive axioms). The constructed system has infinitely long schema.

Basically, I suspect you have wandered off into the realm of "take a sphere, cut it up, rotate, translate, get 2 spheres", in which the truth of your theorum depends on axioms which you can flip-flop either way.

That's Banch-Tarski paradox - a result of application of axiom of choice. I'm not sure if we need it to construct our system.

I don't know about you, but I don't know what it means to have a logic system in which the axioms are provably non-computable to be "complete".

They are not computable in finite time (or without a halting oracle), but this system does exist.

However, it's totally impractical for any purpose except showing me that I was wrong :)

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Postby saibot834 » Thu May 24, 2007 7:25 pm UTC

bbctol wrote:
saibot834 wrote:This comic is very funny :)

Today was my math test (I'm still in school) and I printed out the comic and gave it back along with my test... let's see if my math teacher thinks too, that it's funny and perhaps he will give me some extra points :)


Is there a reason for the word "bot" in your name? Or should I be nervous?

Nah, it's just my real name "tobias" inverted -> saibot. I'm no bot, sorry ;)

Hehe, I think my math teacher laughed about the comic, but he told me that he doesn't think that math is "perfect universal truths", because math is only something humans created and not a "universal truth" itself...

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Postby skeptical scientist » Fri May 25, 2007 4:58 am UTC

Cyberax wrote:
Basically, I suspect you have wandered off into the realm of "take a sphere, cut it up, rotate, translate, get 2 spheres", in which the truth of your theorum depends on axioms which you can flip-flop either way.

That's Banch-Tarski paradox - a result of application of axiom of choice. I'm not sure if we need it to construct our system.

It's "Banach-Tarski", by the way, in case that wasn't just a typo.

When we're talking about the existence of complete extensions of theories, we don't need the axiom of choice or any of the other axioms which you might find suspicious. Complete extensions of theories do exist, and the statements about mathematical logic that you can prove by using the existence of complete extensions of theories are true in standard ZF set theory, without the axiom of choice (i.e. provable from all of the usual axioms mathematicians use, even without using choice).

Statements which are provable in your complete extension of a theory, however, are in general very suspicious and wishy-washy. Unless you started out with a complete theory to begin with, you are making many many choices about the axioms you use, and so you can end up with all sorts of different statements being true or false in the complete extension. But that's not what we use it for - the existence of a complete extension is generally used in such a way that we don't care what the complete extensions is, or what statements are provable from the ridiculously unknowable set of axioms we chose when constructing it. We just care that the fact that complete extensions exist is useful for proving some interesting statements about the theories we actually do work with. In other words, there are proofs that follow the basic structure:
1) Start with a theory we like, such as PA.
2) Extend PA to a complete theory T.
3) Prove some result about T by using, among other things, the fact that it is complete.
4) Show that the result you proved of T implies some interesting fact about PA.
5) Conclude that PA has some interesting previously unknown property.

Yakk, do you buy what we're doing now? Just because a proof requires a step that we can't explicitly construct doesn't mean it's result is false. At least, not unless you agree with Brouwer and think that you have to be able to construct something for it to exist, i.e. that continuous self-maps of closed unit balls may not have fixed points.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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Politicizing math

Postby jwgrendel » Thu May 31, 2007 5:23 pm UTC

I was reading a Dunesbury comic strip a couple years back, and they had a plot line about a college professor being sued by a student for the loss of income that he inflicted by giving him a B+ instead of an A. In a subsequent strip another professor was explaining to a friend how the whole college is going soft on grading now and even he found himself doing it. His friend replies “Wait, you’re a math teacher!â€

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Postby Yakk » Thu May 31, 2007 6:58 pm UTC

skeptical scientist wrote:
Cyberax wrote:
Basically, I suspect you have wandered off into the realm of "take a sphere, cut it up, rotate, translate, get 2 spheres", in which the truth of your theorum depends on axioms which you can flip-flop either way.

That's Banch-Tarski paradox - a result of application of axiom of choice. I'm not sure if we need it to construct our system.

It's "Banach-Tarski", by the way, in case that wasn't just a typo.

When we're talking about the existence of complete extensions of theories, we don't need the axiom of choice or any of the other axioms which you might find suspicious.


I find ~\Forall X in S ~P(X) -> \Exists X in S P(X) to be suspicious myself, when applied to arbitrary set S. :)

1) Start with a theory we like, such as PA.
2) Extend PA to a complete theory T.
3) Prove some result about T by using, among other things, the fact that it is complete.
4) Show that the result you proved of T implies some interesting fact about PA.
5) Conclude that PA has some interesting previously unknown property.

Yakk, do you buy what we're doing now? Just because a proof requires a step that we can't explicitly construct doesn't mean it's result is false. At least, not unless you agree with Brouwer and think that you have to be able to construct something for it to exist, i.e. that continuous self-maps of closed unit balls may not have fixed points.


Some of them have fixed points.

I don't think a theory in which an automorphism of a closed unit ball without a fixed point exists as being more than a play theory: an interesting experiment in what can be consistent.

Insisting that the fixed point exists for any smooth automorphism, however, seems like needless completionism to me. I'd believe that any smooth automorphism takes some point to another point arbitrarially close to itself, if that makes you feel better?

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proof_man
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Re: "Certainty" Discussion

Postby proof_man » Mon Jan 14, 2008 11:18 am UTC

edit
Last edited by proof_man on Fri May 17, 2013 12:56 pm UTC, edited 1 time in total.

jdogmoney
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Re: "Certainty" Discussion

Postby jdogmoney » Mon Feb 04, 2008 1:57 am UTC

Perfect universal truths?


i think not.
Luckily, I have the [noun] of science.

ErrorUncertainty
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JS Mill

Postby ErrorUncertainty » Thu Feb 28, 2008 7:16 pm UTC

The confused teachers would be very much helped by the teachings of John Stuart Mill

He was troubled by exactly this problem, and arrived at this conclusion:

There are no absolute truths, or floating metaphysical 'Forms' that are approached with imperfect truths (Plato), or universal timeless truths.
The only truth we can know is that which we arrive at through discussion and argument with other people.
The teachers can discuss with pupils, peers, lecturers etc. their subject matter, find the best argument and the best expression for it, and it is this that is the 'truth', or as close as said teacher can realistically get.
The teacher then has the duty to bring that truth to his pupils, and should they not agree with it, discuss it with them and either persuade or be persuaded.

I hope everyone can see the magic of this :) i've recently been converted to Mill, and its wonderful as an outlook on life.
Like the one I already held, only more true.

AvalonXQ
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Re: "Certainty" Discussion

Postby AvalonXQ » Thu Feb 28, 2008 7:53 pm UTC

If anybody actually thinks that math imparts truths that are either "universal" or "perfect", they've drank too deeply of the kool-aid.
Math is nothing more than very, very useful grammar taken to extremes.

Lunar Savage
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Re: "Certainty" Discussion

Postby Lunar Savage » Wed Dec 16, 2009 2:04 am UTC

I disagree.

Now, if I'm way off base, discount my post (I'm a tad drunk right now...sooo.... :lol: ).

I do believe math has failed to explain what happens inside of a black hole. So, perfect universal truths? I think not. :P
*Tips top hat, adjusts monocle, and walks away with cane* and yes, that IS Mr. Peanut laying unconscious on the curb.

aprzn123
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Re:

Postby aprzn123 » Wed Apr 25, 2018 1:48 am UTC

Title text wrote:a(b+c)=(ab)+(ac). Politicize that, bitches.

Okay! a, b, and c all represent political parties. Let's say that they are Amocratic, Bepublican, and Creen. This is saying that if you have the same number of all of them, as on the left, the Aemocratic party is worth as much as the others combined. . . .

Terrible jokes aside, let's move on to the other terrible jokes.

Shpadoinkle wrote:
wibblymat wrote:The alt text needs prefixing with "For all a,b,c in a field," or similar. Maths is only as perfect as the way you write it down!


No, math remains perfect. The problem lies in human error and/or the inability to properly understand it.

Stop splitting infinity!* It's not a number, don't use it in division!

*I should have said infinitives, but then the joke wouldn't work.


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