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Not all reasoning is logic. Logic, by virtue of what it is is universal.
Of course if you went on about logic not existing and you imposed your thoughts about how nothing should have structure and how things are flawed you will of course receive criticism of some sort. It’s when you make your thoughts into actions when you will be struck down. I agree with you - nothing in life makes sense at all – but because of 1000s of years of vices that smarter and stronger human beings created the word "civilized" has now been imposed on your being. It doesn’t seem fair that you have to be involved in their justice system but you are. You can do anything you put your mind to but you'll have to deal with others imperfections when you mess with their business.See if your reason stands up to public scrutiny
I tend to agree until personal logic enters the picture. It may seem to be based on universal princibles yet, in the end, it is based on emotion and what makes the person feel good.Not all reasoning is logic. Logic, by virtue of what it is is universal.
If it is based on emotion it is no longer logic, but rather an entirely different mode of reasoning.I tend to agree until personal logic enters the picture. It may seem to be based on universal princibles yet, in the end, it is based on emotion and what makes the person feel good.
Emotions do not coincide with logic.I tend to agree until personal logic enters the picture. It may seem to be based on universal princibles yet, in the end, it is based on emotion and what makes the person feel good.
What do you take proving something to mean? It sounds like you are saying that if you walk from X to Y, you have walked in a circle because there was a time at which you were not walking.Is all logic/reasoning circular?
Consider this: Science uses experimentation and physical evidence (logic, reasoning, and the senses) to prove or disprove a hypothesis, theory, or judgment. The hypothesis/theory/judgment, however, was made by using logic/reasoning/senses.
So you are disturbed by the fact that your conclusion follows from premises? That is what a conclusion is. Perhaps you are expecting a conclusion to be absolute in some way. If so, you are expecting too much from it. You have to start somewhere, or yes, you cannot get anywhere, not even in circles.We essentially "created" human logic and reasoning. By using logic and reasoning to prove theories that were created by our own logic and reasoning, are we really proving anything at all?
No it doesn't. It only applies toWhile the theorem deals with math it applies to all rational constructs.
No, they are said to be incomplete in the sense that there exists a statement in its language that it can neither prove nor disprove.These constructs are said to be consistent with themselves but are incomplete in that they cannot prove themselves.
IMHO, the main contribution of this theorem is that such a thing can be done, so it's worth seeing what else might be incomplete, not that it says anything about anything else.Gödel's incompleteness theorems states
....
At some point, you have to reach axioms. I don't know of any such infinite regression. If you try to force it, you'll just end up restating stuff.My understanding is a logical system can be made complete if it's situated within a context of other system(s). For example, the scientific method heavily depends on mathematics(which depends on other constructs, which also depend on other constructs, etc...) and vice versa. The issue here is it becomes an infinite regression of constructs. So which do you prefer? Circular but consistent or Infinitely regressing but complete?
True enough, however the nature of axioms are universal regardless of what construct they are used in.No it doesn't. It only applies to
consistent formal, computably enumerable theories that prove basic integer arithmetical truths.Those are the hypotheses of the theorem, and thus are the only things that this theorem can prove incomplete.
I'm just an armchair thinker here so you'll have to forgive my lack of in depth experience with either one of those. My understanding is it's impossible to have a construct be both simultaneously complete AND consistent due to the fact that all constructs rest on axioms. It's just the nature of the relationship we have with reality. We can clearly see that 1 + 1 = 2, and use that as an axiom to build up the construct consistently however you can't use that construct to prove that 1 + 1 = 2, since the construct assumes that it's true already. In the end it would be like saying 1 + 1 = 2 therefore 1 + 1 = 2.The elementary theory of Euclidean geometry and the elementary theory of real number arithmetic are notable examples of consistent formal, computably enumerable, complete theories. (In fact, they are essentially the same theory)
A theory can prove it's own consistency, it just can't be complete at the same time.Furthermore, I'm fairly certain that there exist consistent formal theories that can prove their own consistency. (which, of course, requires that the theory fails to include integer arithmetic, or that it fails to be computably enumerable)
True enoughNo, they are said to be incomplete in the sense that there exists a statement in its language that it can neither prove nor disprove.
True, the thing is it applies to all axiomic systems. Basically everything.IMHO, the main contribution of this theorem is that such a thing can be done, so it's worth seeing what else might be incomplete, not that it says anything about anything else.
In a consistent system you end up where you started. What can be done is the axioms can be pawned off to another construct to test it's provability(out of the box so to speak). The problem with this is the other construct itself rests on axioms(whose own axioms can be pawned off to another construct that rests on axioms to infinity).At some point, you have to reach axioms. I don't know of any such infinite regression. If you try to force it, you'll just end up restating stuff.
And I'm telling you that your understanding is incorrect.My understanding is it's impossible to have a construct be both simultaneously complete AND consistent due to the fact that all constructs rest on axioms.
And that's a perfectly valid proof.In the end it would be like saying 1 + 1 = 2 therefore 1 + 1 = 2.
Do you have a reference for that? I can't seem to find one.A theory can prove it's own consistency, it just can't be complete at the same time.
I fail to see how this doesn't rest on unproveable axioms(e.g 1 + 1 = 2).And I'm telling you that your understanding is incorrect.
The content of Tarski's theorem^{1} is that the elementary theory of real arithmetic^{2}, or equivalently the elementary theory of Euclidean geometry, is simultaneously consistent complete, consistent, and computably enumerable.
It's a perfectly valid observation, a perfectly valid observation but you can't "prove" its validity by using mathematics(since mathematics simply assumes it's true). You can only observe it. As such it's proof lays outside of the formal system in direct observation.And that's a perfectly valid proof.
It's just Kantian philosophy. It's not specific to mathematics.Do you have a reference for that? I can't seem to find one.
Interesting stuff, I just may have to try and wrap my head around it.1: alas, there are a few that go by this name, and Wikipedia doesn't have an entry for the one I'm talking about
2: if you want a buzzword to start investigating, try real closed field.
It seems like "proof" is being used in more than one way. To some people, an axiom is a proof of itself, so there is no such thing as an "unproveable axiom".
What do you take "proof" to mean? Logic and mathematics have defined the term for themselves already. You seem to be using a different meaning.
I'm talking about proving somethings existance using a formal system without being circular.
The process of deduction introduces givens that aren't neccasarily proven. Assumptions about our relationship to reality, logic, etc...Also, a deductive system doesn't even need to have any axioms. You can have rules that allow you to deduce sentences from "nothing" (or the empty set).
So you are talking about a formal proof, then? How about the following definition?I'm talking about proving somethings existance using a formal system without being circular.
What is "the process of deduction"? A proof? What is a given? An axiom? Are you again claiming, with different words, that there exist unprovable axioms?The process of deduction introduces givens that aren't neccasarily proven. Assumptions about our relationship to reality, logic, etc...
I'm not talking about a proof. I'm talking about proof as in being able to prove it.So you are talking about a formal proof, then? How about the following definition?
Definition. If L is a formal language and F is a set of L-formulas, a proof of an L-formula f_{n} from F is a non-empty finite sequence of formulas f_{1}, f_{2}, ... f_{n} such that, for each f_{i}, at least one of the following is true:
1) f_{i} is an axiom
2) f_{i} is in F
3) f_{i} follows by your rules from some combination of L-formulas occurring earlier in the sequence.
There's no point in continuing to use words whose meanings aren't agreed on, yes?
A given is something that seems to be readily apparent and is assumed to be true. An axiom is considered a given yes.What is "the process of deduction"? A proof? What is a given? An axiom? Are you again claiming, with different words, that there exist unprovable axioms?
What formal deductive system says anything about reality? Formal deductive systems don't actually "say" anything in the usual sense. If you want to interpret them as making promises that they cannot keep, that sounds like it's not their problem.
I assumed people would be ok with me using the term axioms outside of mathematics. Myth of the given seems to be the more appropriate phrase when talking generally. A deduction begins with a premise which is the given. We can split hairs here but it's essentiallly the same thing.And how is that a response to my comment? You seemed to have problems with axioms, so I thought it might be helpful to point out that you don't necessarily need them.
There's nothing wrong with it, I was just pointing out that it's not neccasarily circular because the givens can then be validated from another perspective(which also rest on givens)And by the bye, what exactly is wrong with being circular in the sense of having your conclusion be one of your premises, besides perhaps usually being a waste of time?
What do you mean by "to prove" if you don't mean "to provide a proof"?I'm not talking about a proof. I'm talking about proof as in being able to prove it.
=PWhat do you mean?
Turn A, B, and C into a real statement, digg down deep enough, and you'll end up where you started.Then by knowing that A is true, then we can know that B and C are true. But knowing A, B, and C is not the clincher that establishes A therefore B, and B therefore C. Ways to make this estabilished (in the sciences) involve making it acceptable by the (scientific) parties involved with the subject. This is done groupwise.
I have nothing against circular reasoning I find it rather useful.An example of circular reasoning is this:
I don't have a job because I don't have any skills.
I don't have any skills because I didn't go to school.
I didn't go to school because I didn't have any money.
I didn't have money because I didn't have a job.
Circular reasoning can be used to describe viscious and virtous circles. They are valid arguments and can describe our natural world, yet they are circular.
So don't think of circular reasoning so negatively. They can and do make logical descriptions of reality. On the other hand, is all reasoning circular?
The problem with that is we never ever directly experience the two balls or the scale. Our experience of the other is always mediated through the mind(culture, constructs, etc..). As such all similar thinking is laid down on assumptions that we hold about our relationship to the outside world. Once you start digging deep enough in the foundations of your reasoning, you start running in circles.Well if there is a strict uni-directional casuality between A and B, then the system can be described without circular reasoning; actually it must not involve circular reasoning. For example:
fact 1: There are only two one pound balls on a given weight scale.
result: There are two pounds on this weight scale.
However, you cannot determine that there are two one pound balls simply by knowing the weight on the scale. Therefore sound circular reasoning concerning this is impossible. Circular reasoning is avoided when one reasons from the specific material to the general material, since you cannot logically deduce something specific (e.g. the two balls each weighing a pound on the scale) from a generality of what exists (e.g. the two pounds on the scale).
I consider them on equal footing.I'm still confused about why you consider the axioms to be any more or less "provable" than the other theorems that you derive from them. All that a proof of Q from P says is that Q follows from P by your rules. From outside, a proof of Q from P doesn't give you Q. It gives you a proof of Q from P.
I don't have a problem with it. I was just discussing the limitations of only being able to observe reality through mediated constructs. That's not saying the rational pov is a mediating construct and the pre-rational pov is not. All of our experience of the outside world is mediated.And why this relation (you can view syntactic and semantic consequence as relations just like any other relations) should bother someone more than other relations (does anyone have a problem with equality?) also escapes me.
Well, outside of the system, you don't get P. You can get a proof of P from P (and so on).How do we get P? And how did we get that? and so on?
Oh. Roger.I don't have a problem with it. I was just discussing the limitations of only being able to observe reality through mediated constructs. That's not saying the rational pov is a mediating construct and the pre-rational pov is not. All of our experience of the outside world is mediated.