Is deductive logic consistent and how does it influence other systems?

[Bob3141592]

Originally posted by selfAdjoint
Tarski constructed a proof at the same level as Goedel's, in the meta theory of geometry, that showed that Geometry was complete. Tarski's proof is just as valid as Goedel's and is well known in the foundations community. Google on Tarski, complete.

Since you are so boulversee by this old news (Tarski's proof is from the 1940s), I won't mention to you the BSS machine and the vast spectrum of complete theories it has opened up.

Gee, if I knew what boulversee meant I could decide if I should feel insultd or not. The closest I could find was bouleversement which means "an overthrow; confusion; convulsion." and that certainly doesn't come across as flattering!

Like I said, I'm not a mathematician by any means, so I'm not in touch with the foundations community. I've read Nagal's book on Godel Proof, and Hofstadter's GED and Klne's The Loss of Certainty. Granted, I might not have fully understood them, and since they ain't textbooks, you might not think them worth much, but I do what I can.

Tarski... have I ever heard of him? Was he the one that proved you could take a sphere apart and then put it back together to make a sphere of twice the volume of the original? Is that included in his geometry?

 

[master_coda]

Originally posted by Canute
But surely an axiomatic system that proves its own axioms is not an axiomatic system. An axiom is by definition not derived from other theorems in the system.

That isn't quite accurate. An axiom is not defined by saying that you cannot derive it from other theorems in the system. Rather, an axiom is just a statement that you have assumed to be true.

You can often show that the axioms are true from the theorems. But since the theorems were derived by assuming the axioms are true, this isn't really meaningful...all you've done is show that if you assume the axioms are true, then you can prove the axioms are true.


What Tarski did was to provide a set of axioms for Euclidean geometry, and show that given any geometric statement G, you can either:
1. Derive G from the axioms.
2. Derive the negation of G from the axioms.

Obviously you can also derive the axioms from the axioms, since A => A is a tautology. But Tarski wasn't trying to show that his axioms were "true". He was trying to show that in Euclidean geometry, there are no questions that "cannot be answered".

 

[master_coda]

Originally posted by Bob3141592
Tarski... have I ever heard of him? Was he the one that proved you could take a sphere apart and then put it back together to make a sphere of twice the volume of the original? Is that included in his geometry?

This is the Banach-Tarski paradox. Given a ball in R^3, you can divide the ball into six pieces, and re-assemble them to form two balls of the same size as the original.

However, this paradox depends upon the axiom of choice. That is not an axiom of Euclidean geometry.

 

[Hurkyl]

None of the axioms of set theory, incidentally, are part of Euclidean geometry. (Though you can simulate some very simple set things via logic)

 

[selfAdjoint]

I didn't mean bouleversee in an insulting way, but only to jolly you along a bit, because you did seem upset that the Goedel theorem didn't have the universal scope you had ascribed to it.

There is active work going on now to see how much of the computations with real numbers can be incorporated into a complete system. The BSS machine I mentioned is one of the (purely abstract) tools for doing this.

By the way, I have read Nagel and Hoffsteader too. I can never decide if I like the Nagel book or hate it. About Hoffsteader I am ravingly enthusiastic.

 

[Bob3141592]

Originally posted by selfAdjoint
I didn't mean bouleversee in an insulting way, but only to jolly you along a bit, because you did seem upset that the Goedel theorem didn't have the universal scope you had ascribed to it.

No, not upset. I'd rather learn somethin than go on being wrong. I know just enough about these things to be dangerous, so I try to never trust what I think I know.

By the way, I have read Nagel and Hoffsteader too. I can never decide if I like the Nagel book or hate it.

Sounds ironically appropriate, doesn't it?

About Hoffsteader I am ravingly enthusiastic.

I've bought four copies of Hofstadter's book, since I'm always loaning it out and don't always see it returned. Not often you find a mathematics book with real wit and humor in it.

I tell people even if they find the body of the sections to be too much, just read the dialogues. Generally, if someone actually goes that far, they ask to keep the book longer so they can read the whole thing.

By the way, I knew my comment about Tarski wasn't legit. I was trying to be snide back, in response to the perceived slight. Sorry about that.

 

[Canute]

I suspect we're a bit at cross purposes here. Bob brought up the issue of whether reality itself forms an axiomatic system (which I take to mean - can be fully described by such a system). A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent. This raises a few interesting question about reality and our ability to reason our way to the truth about it.

Whatever Tarski proved is not relevant here, because Tarski's axiomatisation of geometry works precisely because his system is entirely self-referential, it does not refer to anything real. As Einstein said somewhere "So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”

This is what I meant by saying that Goedel's theoerems are not in any way limited in their applicability by Tarski's work. They are not dealing with the same kind of systems. Goedel's theorems simply don't apply to systems that are completely circular.

I'd like to discuss this because I've been trying to sort out some issues for quite a while but this is the first time I've found myself talking to people who seem to know something about these things.

My interest is in the implications of Goedel for science and metaphysics rather than for mathematics. Dangerous territory but fun to explore.

Hawkings says this:

"What is the relation between Goedels theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted….( )

In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed, and can contain an arbitrary amount of information, without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Goedels theorem. One might therefore expect it to be either inconsistent, or incomplete. The theories we have so far, are ~both inconsistent, and incomplete."

(Stephen Hawking – Goedel and The End of Physics – net article (http://www.damtp.cam.ac.uk/strtst/dirac/hawking/ )

The question for me is whether the incompleteness theorems are just a quirk of epistemology or whether they tell us something important about reality. I feel it is the latter. They apply in all possible universes, and seem to place limits on what can be known by any system of proofs. Even if we take our perceptions as axiomatic (I think therefore I am) the same problems arise. (A fact I believe Plato saw, hence the shadows on the cave wall and our inability to reason our way out of the cave).

I feel that this relates directly to the use of undefined terms.

“…since every word in a dictionary is defined in terms of another word…The only way to avoid circular reasoning is a finite language would be to include some undefined terms in the dictionary. Today we must realize that mathematical systems too, must include undefined terms, and seek to include the minimum number necessary for the system to make sense.”

Leonard Mlodinow – Euclid’s Window p144

All this suggests that to explain reality one must have, at minimum, three ingredients in the explanation, namely an axiom, an undecidable question and an undefined term. Then one has to circumvent Goedel.

I'd like to hear your comments on this because it's such a slippery topic, expecially for a non-mathematician, that know I might be misunderstanding some of the issues.

I feel that the incompleteness theorems can be understood as arising from the nature of reality, if one assumes that it arises from a 'non-dual' ontology. (Something Hofstaedter just missed seeing in GEB, although I don't know how, he was so close). But before trying that one out I'll wait for a response to this bit.

 

[Bob3141592]

Originally posted by Canute
I suspect we're a bit at cross purposes here. Bob brought up the issue of whether reality itself forms an axiomatic system (which I take to mean - can be fully described by such a system). A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent. This raises a few interesting question about reality and our ability to reason our way to the truth about it.

I'm not sure that necessarilly follows.

Perhaps much of the turmoil here is caused by a subtly of what Goedel's Proof is really about. The focus of the proof concerns only second order predicate logic. I'm not sure of the details of the distinctions, so it's time to do a bit more work. Here's one place to start -- http://en.wikipedia.org/wiki/First-order_predicate_calculus

Now, back to your original point. Let's say we have a complete and perfect model of reality at the atomic level. Just because the model is abstract and not made of the same "stuff" as the thing it models doesn't mean it's referring to something outside itself. That was never a formal requirement, was it? So I'm not sure this kind of argument applies. Besides, we don't know what the stuff of the universe is actually made of. Perhaps by time they work out all the details they'll find that strings or membranes or whatever is really vibrating in those higher dimensions are really just numbers after all. So, philosophically speaking, even if it mattered, we don't know that the model has to refer to something outside of itself.

Do I think that's really the case? No, not really, but I don't know, and it's not a legitimate assumption one way or another.

Anyway, got to go fix breakfast, and in between chores read up on predicate calculus of various orders.

 

[master_coda]

Originally posted by Canute
A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent.

I just think I should mention this again. The incompleteness theorem does not say anything about systems that refer to things outside themselves. It talks mathematical systems, and no mathematical system refers to anything outside of itself.

If you insist that a system describing reality must refer outside of itself, then you have already shown that we cannot use a mathematical system to describe reality.


I tend to think that incompleteness is more of a property of logic than the universe. Of course, if logic is an intrinsic part of the universe, then incompleteness is also a property of the univserse that we cannot escape. However, I'm not familiar with the philosophical arguments of that form...I'm just a mathematician.

 

[Canute]

Originally posted by Bob3141592
I'm not sure that necessarilly follows.

Perhaps much of the turmoil here is caused by a subtly of what Goedel's Proof is really about. The focus of the proof concerns only second order predicate logic. I'm not sure of the details of the distinctions, so it's time to do a bit more work. Here's one place to start -- http://en.wikipedia.org/wiki/First-order_predicate_calculus

I think it is generally accepted that Goedel's theorems have implications well beyond mathematics. In this I have Roger Penrose and Stephem Hawking on my side.

Now, back to your original point. Let's say we have a complete and perfect model of reality at the atomic level. Just because the model is abstract and not made of the same "stuff" as the thing it models doesn't mean it's referring to something outside itself.

If the model is a model of something else how can it not be referring to that something else. A model refers to something else by defintion. (Are we using 'refer' in different ways?)

So, philosophically speaking, even if it mattered, we don't know that the model has to refer to something outside of itself.

I think we do. What's the point of a model of reality that doesn't refer to anything?

An axiomatic system can be defined as a system that refers outside of itself, since its fundamental axiom is an theorem about something other than the system. It points outwards rather than inwards. (E'g' I think therefore I am, there is a line such that ... , God exists, etc).

 

[Canute]

Originally posted by master_coda
I just think I should mention this again. The incompleteness theorem does not say anything about systems that refer to things outside themselves. It talks mathematical systems, and no mathematical system refers to anything outside of itself.

I don't agree I'm afraid. The incompleteness theorems apply to axiomatic systems. Axioms refer outside the system by definition.

Also mathematical systems can refer ourside themslves. But, as Einstein said:

“ So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”.

If you insist that a system describing reality must refer outside of itself, then you have already shown that we cannot use a mathematical system to describe reality.

I think that there's a get out clause.

I tend to think that incompleteness is more of a property of logic than the universe. Of course, if logic is an intrinsic part of the universe, then incompleteness is also a property of the univserse that we cannot escape. However, I'm not familiar with the philosophical arguments of that form...I'm just a mathematician. [/B]

I'm pleased to be talking to a mathematician about this, since it keeps some rigour in the discussion. My maths is rubbish I'm afraid, but to me Goedel has far more significance outside mathematics than inside it. If you're interested in the philosophical side and haven't already read it Penroses's stuff on this is brilliant, (since he can cope with the maths!) although many believe he went too far with his metaphysics in almost proving that God exists.

 

[master_coda]

Originally posted by Canute
I don't agree I'm afraid. The incompleteness theorems apply to axiomatic systems. Axioms refer outside the system by definition.

Also mathematical systems can refer ourside themslves. But, as Einstein said:

“ So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”.

How do axioms refer to something outside of the system?

For example, in set theory one of the standard axioms is the axiom of the empty set. The axiom asserts that there exists a set that contains no elements.

But that does not refer to anything outside of the system (set theory). It is considered to be an axiom because it is asserted to be true instead of being deduced to be true.


The Einstein quote is a common one...it's usually used to try and put mathematical physicists in their place. But although the quote says "laws of mathematics refer to reality", that isn't really accurate. The math itself does not refer to reality in any way.

The quote is actually referring to the practice of taking mathematics and trying to derive conclusions about reality from math. Just because one can show that Euclidean geometry is consistent, it doesn't mean the nature follows Euclidean geometry.

 

[Canute]

Originally posted by master_coda
How do axioms refer to something outside of the system?

For example, in set theory one of the standard axioms is the axiom of the empty set. The axiom asserts that there exists a set that contains no elements.

But that does not refer to anything outside of the system (set theory). It is considered to be an axiom because it is asserted to be true instead of being deduced to be true.

But doesn't it assert something about reality that is not part of the system? Isn't that precisely what makes it an axiom?

The Einstein quote is a common one...it's usually used to try and put mathematical physicists in their place. But although the quote says "laws of mathematics refer to reality", that isn't really accurate. The math itself does not refer to reality in any way.

As a mathematician you may be forgetting that people use mathematics to count their change, not just for developing heuristic systems of proof.

The quote is actually referring to the practice of taking mathematics and trying to derive conclusions about reality from math. Just because one can show that Euclidean geometry is consistent, it doesn't mean the nature follows Euclidean geometry. [/B]

I agree. If one can show that Euclidean geometry is consistent then one can know it says nothing about reality, as Einstein suggests.

If it is to say something about reality, to prove something about it, a system cannot be made entirely self-referential. It seems to me that at some point it has to make an assertion that cannot be deduced but must be checked against the external facts. Is this not so?

 

[Hurkyl]

But doesn't it assert something about reality that is not part of the system? Isn't that precisely what makes it an axiom?

No. It's an axiom merely because it was selected to be an axiom.



You're missing a major piece of the puzzle here. It is not the job of a mathematical system to say anything about reality. It is someone else's job (e.g. a physicist) to devise a correspondence between reality and a mathematical system. It is through this correspondence (which is not part of the system) that the system says anything about reality.

 

[Canute]

Originally posted by Hurkyl
No. It's an axiom merely because it was selected to be an axiom.

I doubt that you can give an example of a fundamental axiom which refers only to the the system of theorems which can be derived from it.

Here's Anton Setzer from http://www-logic.stanford.edu/proofsurvey.html

"I think proof theory is mainly dealing with foundations, and after some mathematical reduction steps we will always end up with some principles, which can only be validated by philosophical arguments. Here interaction with philosophy is required.

You're missing a major piece of the puzzle here. It is not the job of a mathematical system to say anything about reality. It is someone else's job (e.g. a physicist) to devise a correspondence between reality and a mathematical system. It is through this correspondence (which is not part of the system) that the system says anything about reality. [/B]

I don't think it matters whose job is what. A mathematical system must be axiomatic in structure (based on or containing an underived/unproved theorem) in order to refer beyond itself, just as a dictionary must contain an undefined term in order to do so. That doesn't mean mathematical systems have to refer to anything, many of them don't, as you say.

 

[Hurkyl]

I doubt that you can give an example of a fundamental axiom which refers only to the the system of theorems which can be derived from it.

I'm not entirely sure what you're trying to imply, so I'll take a stab in the dark; how about this axiom of real arithmetic:

$$\forall a,b \in \mathbb{R}: a + b = b + a$$




And as I mentioned before, any connection between reality and a mathematical system is not part of the mathematical system.

 

[Bob3141592]

Originally posted by Canute
I don't agree I'm afraid. The incompleteness theorems apply to axiomatic systems. Axioms refer outside the system by definition.

Also mathematical systems can refer ourside themslves.

Ah, I think this is where the discussion diverges.

A formal mathematical system means that the axioms are abstract statements containing various symbols that are assummed to be true. The sysmbols used in the axioms don't actually have any meaning. As such, they refer to nothing.

When a mathematician writes F = ma, he is making no statement beyond how vectors multiply together. But when you say force equals mass times accelleration, then you have added meaning to the symbols that are not part of the mathematics. Sure, they look the same, but even in this very simple equation of physics, you use a variable "m" to stand for something, mass, that in reality isn't really understood. The whole thing is more than just an approximation, it's a simplification that intentionally ignores major parts of reality in order to shoehorn it into a mathematical framework.

It's quite astounding that it seems like the complicating factors that were simplified away can be accounted for, and added back in by physicists. It takes trained physicists, because the equations get more and more involved, and more complicated, and most people go absolutely apoplectic when they see them. But no matter how elaborate those equations of physics become, they continue to make simplifying assumptions, and to leave out complicating factors tat make working with the equations just too complicated. Nobody even tries to write equations even for something as simple as an Estes model rocket using quantum mechanics. But if they don't, they can't pretend that the equations they use are an exact, meaningful represetation of reality. Most physicists don't even try to make that claim. It's only an approximation, and generally, they're quite happy that they get the aproximation good enough that it can be scaled up to throw bigger rockets at Mars. They're esctatic when their aim is true, but they don't think of it as Truth.

I suppose there is a class of physicists who are looking for Truth. t'Hooft wants to find the "ultimate" building blocks. Lederman wanted to find the "God particle" (though I understand it was more often referred to as the "Goddam particle." But it's presumptuous of us to asume we understand how those people really think philosophically. It's more that if they didn't write poeticaly, people like us wouldn't be able to understand anything they said.

 

[Bob3141592]

Originally posted by Hurkyl
I'm not entirely sure what you're trying to imply, so I'll take a stab in the dark; how about this axiom of real arithmetic:

$$\forall a,b \in \mathbb{R}: a + b = b + a$$


And as I mentioned before, any connection between reality and a mathematical system is not part of the mathematical system.

Ironic, isn't it, that the example you gave to show mathematics isn't equivalenty to reality says "for everything in the Reals

Perhaps a more philosophically poetic example would involve the imaginary numbers instead of the reals.

And it remains ironic that physics requires imaginary operations to produce "real" results, though most casual philosophers I've talked to think imaginary numbers are entirely unreal and, well, imaginary.

 

[Hurkyl]

Bah fine. How about the ring axiom:

$$\forall a, b: a + b = b + a$$

 

[selfAdjoint]

Originally posted by Hurkyl
Bah fine. How about the ring axiom:

$$\forall a, b: a + b = b + a$$

That's a commutative ring axiom (or an Abeliean group axiom!)

I am not sure I understand what discussing individual axioms is getting us in this discussion?

 

[Canute]

I wonder if this disagreement, my part of it anyway, is caused by my use of language, which may be technically sloppy, me not being a mathematician n'all.

All I'm defending is the idea that an axiom is an assumption, that by defintion it is not derived from the system for which it is an axiom, and that it therefore refers to something beyond that system. I'm still not clear how there can be any objection to this.

If I'm wrong then fine, my world-view doesn't depend on it. But I haven't seen anything I regard as a counter-argument yet. To be honest to me it seems inarguable.

I suppose, in Kantian terms, I'm arguing that systems with no axioms (or axioms that are provable within the system) are analytic, (self-referential) whereas systems with axioms are synthetic (refer beyond themselves).

Or, in logical positivist terms, that complete systems, in which the axioms are provable, are by definition tautological and thus trivial (in a formal sense).

Thus if an axiom is provable (all red roses are red) it can never give rise to knowledge of anything beyond the system. But if it is not (all roses are red) then it can, because it makes a claim about something outside the system.

 

[Hurkyl]

As far as I can tell, your idea of an axiom is simply not the mathematical idea of an axiom.

 

[Bob3141592]

Originally posted by Canute
I wonder if this disagreement, my part of it anyway, is caused by my use of language, which may be technically sloppy, me not being a mathematician n'all.

All I'm defending is the idea that an axiom is an assumption, that by defintion it is not derived from the system for which it is an axiom, and that it therefore refers to something beyond that system. I'm still not clear how there can be any objection to this.

If I'm wrong then fine, my world-view doesn't depend on it. But I haven't seen anything I regard as a counter-argument yet. To be honest to me it seems inarguable.

Well, the first part of that is right, an axiom is an assumption and it isn't derived from the system. The second part, the conclusion, is wrong. An axiom does not refer to something beyond the system. The axiom is one of the foundation parts of the system; there is no system without the axioms. And that's a big objection to what you're describing.

Perhaps what you mean isn't an "axiom" but an "observation." Models built to explain observations can use axiomatic systems in their method of explanation, but that isn't equivalwent to the model.

So perhaps it's all a matter of terminology.

I suppose, in Kantian terms, I'm arguing that systems with no axioms (or axioms that are provable within the system) are analytic, (self-referential) whereas systems with axioms are synthetic (refer beyond themselves).

What system has no axioms? That doesn't make any sense to me. Axioms are a crucial part of the definition of the system.

Or, in logical positivist terms, that complete systems, in which the axioms are provable, are by definition tautological and thus trivial (in a formal sense).

Axioms don't need to be proved. It's a given that they're true (at least, within the system they help define). If a system proves it's axioms false, then the system is contradictory.

I'm not certain about this, but I think even inconsistent systems don't prove their axioms false. An inconsistent system allows derived statements to be shown as both true and false. But that applies to derived statements, not the axioms. Is that right?

Thus if an axiom is provable (all red roses are red) it can never give rise to knowledge of anything beyond the system. But if it is not (all roses are red) then it can, because it makes a claim about something outside the system.

I don't think any formal system can ever give rise to knowledge of anything beyond the system. The symbols used in formal systems are in and of themselves meaningless, but the way you're talking it seems you assume they have a meaning that applies to other things.

If you're trying to build a model of other things based on observations, and the model develops inconsistencies or contradictions, then the first place to look for problems would be in the observation or the interpretation of the observation. There are countless more ways to go wrong there. You can't asume the observation is perfect and that problems come about because the axiomatic system used to model the observation is defective.

Or do I misunderstand the intent of your argument?

 

[Canute]

Originally posted by Bob3141592
Well, the first part of that is right, an axiom is an assumption and it isn't derived from the system. The second part, the conclusion, is wrong. An axiom does not refer to something beyond the system. The axiom is one of the foundation parts of the system; there is no system without the axioms. And that's a big objection to what you're describing.

I think you misunderstand what I mean. Of course there is no system without the axioms. But the axioms refer beyond the system. Unlike all the other theorems of the system they deal with the outside world, they refer directly to it. That is why they are axioms. If they didn't do this they couldn't be axioms. Instead they would be theorems derived from the system.

Perhaps what you mean isn't an "axiom" but an "observation." Models built to explain observations can use axiomatic systems in their method of explanation, but that isn't equivalwent to the model.

No, that isn't it.

What system has no axioms? That doesn't make any sense to me. Axioms are a crucial part of the definition of the system.

If an axiom is proved within the system then it is not an axiom. If a system proves its own axioms it ceases to be a non-trivial axiomatic system. It becomes entirely self-referential. A system that says that a=b, b=c, c=a is trivial in a mathematical sense, and by one point of view does not even contain an axiom.

Axioms don't need to be proved. It's a given that they're true (at least, within the system they help define). If a system proves it's axioms false, then the system is contradictory.

OK.

I'm not certain about this, but I think even inconsistent systems don't prove their axioms false. An inconsistent system allows derived statements to be shown as both true and false. But that applies to derived statements, not the axioms. Is that right?

Hmm. I think maybe you're right in 'local' mathematical terms, but otherwise I'm arguing that Goedel is right precisely because all axiomatic systems do have false or inconsistent axioms inasmuch as they refer to reality. That's a personal view though, and based on metaphysics more than mathematics.

I don't think any formal system can ever give rise to knowledge of anything beyond the system. The symbols used in formal systems are in and of themselves meaningless, but the way you're talking it seems you assume they have a meaning that applies to other things.

No, I agree with you as far as mathematics goes. But in theorising about reality we have to give those symbols meaning, and interpret what the mathematical behaviour of the symbols might mean, as a physicist would.

If you're trying to build a model of other things based on observations, and the model develops inconsistencies or contradictions, then the first place to look for problems would be in the observation or the interpretation of the observation. There are countless more ways to go wrong there. You can't asume the observation is perfect and that problems come about because the axiomatic system used to model the observation is defective.

Agreed

Or do I misunderstand the intent of your argument?

I think you do, but I can't tell whose fault it is. I may be expressing myself badly, but I'm in the worst position to judge.

 

[phoenixthoth]

a bit of delicacy. some axioms in set theory can be proven from others and they do not need to be called axioms; they could be called theorems. however, there do seem to be axioms that one can prove are "independent" and my understanding is that the gist of that means that one cannot prove the other.

part of what you wrote is that axioms are assumptions and this is my understanding of axioms. i hope this isn't what hurkyl was objecting to because that means I'm wrong. that's what i think the kernel of an axiom is. in addition, it's also a statement along the lines of "this is what our system depends on" and "if you accept this then these must follow."

so instead of being like a=b, b=c, and c=a for those axioms which can prove each other, it's more like a=b+d, b=c, d=c-b. you still have a=b=c yet 1. there is this independent d and 2. these "equations" have more "information" than a=b=c. I'm probably not explaining this right.

 

[Canute]

Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

 

[phoenixthoth]

i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

hmm... what are we really talking about here? read that one more time and you'll get the allusion.

 

[master_coda]

Originally posted by Canute
Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

What outside of the system does that axiom refer to?

 

[phoenixthoth]

math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

 

[master_coda]

Originally posted by phoenixthoth
math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

You cited a number of examples of axioms that "correspond" to what we see the real world. But it is in no way necessary for axioms to manifest themselves some way in reality.

For example, non-Euclidean geometry generally replaces the parallel lines axiom with an axiom that is contradictory to the parallel lines axioms. And non-Euclidean geometry is just as much of a valid system as Euclidean geometry.

Most of the other examples are very circular...they're concepts that relate to the real world because humans use them...and we use them because they relate to the real world. Perhaps they only seem to be "intrisically true" to us because we were raised to believe that.


Whether or not the axioms are "true" outside the system is irrelevant. It's usually best to start with axioms that intuitively seem to be true, but we can just as easily do the opposite, and the math is just as valid.


Of course, using the mathematical definitions of what a formal system is and what axioms are tends to rob those concepts of any philosophical meaning...but if you want to use the incompleteness theorems then you have to use the mathematical definitions. You can't just discard those definitions because you prefer more meaningful ones and keep the theorems at the same time.

 

[phoenixthoth]

part of what i said earlier was:
an axiom only refers to the system it is part of the foundation of (this is what you said in more words). that's what it refers to but that is [edit: usually] not why it exists, ie, why it was created.

as you said, the new axioms from post 1850 or so had no correspondance to reality except that what we imagine is real to an extent in that we really imagine. so those axioms still corresponded to or were invented by an observation of an aspect of reality: the reality inside our minds which of course is not real in the usual sense. but then it turned out that certain axiomatic adjustments lead to reality anyway, which is kind of interesting. i wonder what more we can piece together by adjusting the way we think.

this leads back to max tegmark's TOE article where existence is equivalent to freedom from contradiction.

 

[Canute]

But a non-trivial axiomatic system cannot prove its own axioms. They are only partly part of the system.

(On the problem of explaining the cosmos) – “Every proof must proceed from premisses; the proof as such, that is to say the derivation from the premisses, can therefore never finally prove the truth of any conclusion, but only show that the conclusion must be true provided the premisses are true. If we were to demand that the premisses should be proved in their turn, the question of truth would only be shifted back by another step to a new set of premisses, and so on to infinity.”

Karl Popper – The Problem of Induction (1953, 1974) from ( http://www.dieoff.org/page126.htm)

Aristotle got around this by saying that there are earlier premisses which are indubitably true, and which do not need any proof. He called these ‘basic premisses’.

For both Aristotle and Pooper the problem was the in principle impossibility of proving ones axioms, or basic premisses, without rendering the system trivial.

Inasmuch as human knowledge is based on axiomatic reasoning and proofs Popper also writes this:

“What we should do, I suggest, is give up the idea of ultimate sources of knowledge. And admit that all human knowledge is human: that it is mixed with our errors, our prejudices, our dreams, and our hopes: that all we can do is grope for truth evn though it be beyond our reach. We may admit that our groping is often inspired, but we must be on our guard against the belief, however deeply felt, that our inspiration carries any authority, divine or otherwise. If we thus admit that there is no authority beyond the reach of criticism to be found within the whole province of our knowledge, however far it may have penetrated into the unknown, then we can retain, without danger, the idea that truth is beyond human authority. And we must retain it. For without this idea there can be no objective standards of enquiry; no criticism of our conjectures; no groping for the unknown; no quest for knowledge.”

Karl Popper – ibid.

I believe he was right given his scientific definition of knowledge, but wrong because he defined knowledge too narrowly.

Descartes recognised the problem, and thus chose a fundamental axiom that was in principle impossible to prove.

 

[master_coda]

Of course we can prove the axioms of a non-trivial system. They're true by definition.

I'm well aware of the fact that you can't prove that the axioms are true "in reality". That's why mathematical axioms don't refer to reality. If they did, we would have to justify the axioms (we would have to show that they in fact do refer to the real reality). But since they don't we can just say "define these axioms as true" and they are.

 

[phoenixthoth]

one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

 

[master_coda]

Originally posted by phoenixthoth
one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

I'm not very familiar with the details of forcing. But it isn't for proving consistency, it's for proving relative consistency.

Paul Cohen originally used it to show that if ZF set theory is consistent, then ZF set theory + the negation of the axiom of choice is consistent.