Godel’s First Incompleteness Theorem. Any adequate axiomatizable theory is incomplete. Â In particular the sentence “This sentence is not provable” is true but not provable in the theory.
Godel’s Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of “formula”, “sentence”, “proof”) the consistency of the system in not provable in the system.
If no system is able to prove itself, does that touch on Aristotle’s teaching on “superior sciences”? Each dependant science has principles proven by a superior science.
Since logic is not a science, that gives you your answer. The incompleteness theorem means no single axiomatic system can derive all true statements, or, putting it another way, for any given representation system, true statements can be expressed that cannot be proved by the axiom set of the system.
an interresting case of the irrelevant collation. what you quote about goedel is correct, and interesting. what you provide from yourself is not very readable, not very interesting, and it is not all that obviously related. other than that… what are you asking ?
best i can come up with as a response is pointing out that goedel does not necesarily envisage a hierarchy, while aristotle does not apparently conceive anything else.
Mr. Patterson: “Since logic is not a science, that gives you your answer”
Would you explain to me how logic is not a science? (Scientia being “knowledge”). It’s a knowledge of producing right conclusions, isn’t it? A science of knowledge. (Or does that make it an art?)
Mr. Zenofeller: “an interresting case of the irrelevant collation. what you quote about goedel is correct, and interesting. what you provide from yourself is not very readable, not very interesting, and it is not all that obviously related. other than that… what are you asking ?”
best i can come up with as a response is pointing out that goedel does not necesarily envisage a hierarchy, while aristotle does not apparently conceive anything else."
I am trying to bring forward Aristotle’s doctrine of superior scineces as a response to Godel’s Theorem. If they are put together, i think there is a great deal of science that can be done in a subject once higher sciences’ issues are dealt with. Godel’s by itself would seem to limit every topic to uncertainty since it cannot prove itself. Godel gives us scineces with a hole in them – I wonder if we could plug those holes with references to superior sciences. (Or was this obvious?)
Perhaps things would be easier if you said which between the two you think was the “superior science”, and also explain what exactly you mean by “metaphysics”.
First, is it clear and true that Godel’s Theorem is meant to apply in philosophy as well as math?
Thanks, Dunamis.
Well, I thought metaphysics (study of being) applied to every other study (types of being). Logic deals with all types of being too, which is what confuses me. I guess logic is prior in coming to know, and metaphysics is prior in actually being. You need logic to know metaphysics, but metaphysics explains logic. What a happy mess!
It seems that you have a Phenomenological view of metaphysics, which is fine, as long as we know where you are coming from (if I am wrong, let me know). Perhaps it is best to look directly at the text in Aristotle. In Metaphysics 982a he says,
“Since we are investigating this kind of knowledge, we must consider what these causes and principles are whose knowledge is Wisdom. Perhaps it will be clearer if we take the opinions which we hold about the wise man. We consider first, then, that the wise man knows all things, so far as it is possible, without having knowledge of every one of them individually; next, that the wise man is he who can comprehend difficult things, such as are not easy for human comprehension (for sense-perception, being common to all, is easy, and has nothing to do with Wisdom); and further that in every branch of knowledge a man is wiser in proportion as he is more accurately informed and better able to expound the causes. Again among the sciences we consider that that science which is desirable in itself and for the sake of knowledge is more nearly Wisdom than that which is desirable for its results, and that the superior is more nearly Wisdom than the subsidiary; for the wise man should give orders, not receive them; nor should he obey others, but the less wise should obey him.”
I have italicized what seem relevant points to me. If you are going to argue that metaphysics gives access to knowledge of Being, this would seem to me to be the “ruling science”, but that is a big “if”. It is interesting how it is phrased. Proximity to “sophos” is what grants the superiority of knowledge. The substrate serves the archon science. In Aristotle’s imagery, lying below does not give precedence.
I’m very surprised at you. Your sign quotes Aquinas, and Aquinas spends an entire question on why theology is a science (in Scholastic terms) in I:I of the Summa Theologiae, that explains just what you need to know to derive the answer to your question.
A science is a systematic body of knowledge. Logic is not a systematic body of knowledge; it is a sub-branch of mathematics, which is not a science, either.
There are many ways you could characterize the relationship that mathematics, logic, and linguistics share; my favorite is to characterize them as studies in symbology.
Goedel’s Theorem applies to any system that can be described mathematically, so appealing to it in philosophy would be unexceptionable.
I think MRN specified that he was using the terminology developed by Aristotle in the opening books of the Metaphysics. In thinking through the problem of what was the “first” or most primary science, Aristotle says that disciplines of the arts and sciences derive their principles from more fundamental disciplines – as zoology, for example, derives its principles from biology and biology derives those princnipals which are not unique to biology from physics, and so on. He invented metaphysics, which he called simply “first philosophy” to study the principles of physics – principles more fundamental than those used by physics. Since this exploration followed his writings on physics, it was meta- (after)physics, hence the title of the book and the derived title of the discipline.
Dunamis, I am impressed both that you have a copy of Aristotle’s Metaphysics and that you quoted such a large amount of it!
It’s odd you should say that. I’ve been told before that I think like a Phenomenologist – what’s really strange is that I hadn’t read much if any Phenomenology! Am I going to “the things themselves”? (That’s all I know.)
Mathematics is the science of quantities. I do not see how logic could be part of mathematics, since you need logic to do mathematics.
Here we come to the point where the infinite debate starts, but still, just for fun’s sake:
Are we on meta-physics now about the quality-quantity thing? i just wrote some rant on the “question on kant” thread.
Maths can deal with substances as well. Guess why? Because substances is all about quantities. Ok, emotions. Anger is the result of the acculmilation of agression and repression. Acculmilation, sounds mathematical enough?
no, not metaphysics. i think it’s what’s called “second intention” – thoughts about thoughts.
where do I find this latest “question on kant” thread? is it one of mine?
I thought anger was a detestation of the will, the result of moral outrage. Can you explain a certain Achilles? Sure anger builds up, but why does it build up? How does it help knowledge to know it is a quantity?
You would sorely impress me if you could explain to me a man in terms of lines and numbers.
Okay, Pureasonist, if we assume everything can be known as math, and if some forms of math explain other forms of math, what are the first and final forms of math to be explained or to do explaining?
still, I ask you to reconsider quality and quantity and their relationship.
Man, fundamentally atoms. Dogs, atoms. Trees, atoms. By the time we can fully mathematicalise all these creatures of nature in terms of atoms, we are able to make any of them, via a machine that can combine atoms.
In pure geometry terms, man is in 3 dimensions, fundamentally dots, then lines and planes. We shouldn’t constrain ourselves in the cage of concepts, they are our languages’ creations anyway.
Philosophy is a transcendetal matter (spare me for now arendt ), you can’t talk about one thing without mentioning the other.
Happiness is the aculmilation of satisfaction. When you are satisfied to a degree, you consider yourself as being happy. I’m glad you noticed my signature, it might sounds a little niave to a lot of people, but I believe this is one of the most important things that I’ve learnt from philosophy, via going around and around, eventually to what’s so simple, yet so true and undeniable. Ops, off topic again.
Screw this topic, it’s another philosophical pain anyway. Nice chating with you though.