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Dec
26
revised Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
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Dec
26
comment Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
For a newer discussion check out: New Directions in the Philosophy of Mathematics: An Anthology (Revised and Expanded Edition) by Thomas Tymoczko
Dec
26
comment Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
Checkout metamath and state of the art Automated proving. There are some minor proofs given by computers. This goes back to AI, because if a computer can do mathematics based on axioms, why not program Hilbert's program in a computer. Hofstadter tried to answer this in Gödel, Escher, Bach. Wittgenstein gave a different view on logical atomism. None of this is in the standard curriculum of mathematics, because that would go against the prevailing paradigm. And so everything is build around a certain view.
Dec
26
comment Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
This is a vast subject and in my opinion Bourbaki and mathematical logic is purely formal and there is little to be learned by this exercise. Hilbert's program was put to rest by Gödel more or less. I would start here: en.wikipedia.org/wiki/Introduction_to_Mathematical_Philosophy and here plato.stanford.edu/entries/logical-atomism, and here plato.stanford.edu/entries/frege, and here en.wikipedia.org/wiki/Automated_theorem_proving. Read a book on elementary logic by Quine for example and compare with the questions asked by Russell, Frege.
Dec
25
revised Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
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Dec
25
answered Book on the Rigorous Foundations of Mathematics- Logic and Set Theory
Dec
11
comment What is a Complex Name?
Usually plato.stanford.edu is a good resource, see plato.stanford.edu/entries/descriptions for example.
Dec
11
comment What is a Complex Name?
Metta World Peace, I have tried to do exactly that, i.e. describe the idea of logical analysis. The best thing is to read Frege first, because Russell uses some of his concepts. The most elementary description is his talk on function and concept, see en.wikipedia.org/wiki/Function_and_Concept There are perhaps hundred on textbooks on Frege and Russell, but the best known are two major books of Michael Dummet (philosophy of Language and philosophy of mathematics).
Dec
11
comment What is a Complex Name?
If I have a proof, it is expected from me that people who are in the field can understand it and verify it. But if we are talking about logic, there is no such knowledge which can be assumed. Using abstract algebra to prove an elementary theorem in logic is just non-sensical. We are talking about the most fundamental notions, such as those in predicate logic. And anyone who uses complicated language to express elementary concepts is misusing the language. This is certainly true in this case.
Dec
11
comment What is a Complex Name?
This could be called the elitist belief of mathematicians, that their science has to be some kind of black art. However, often, when pressed with a difficult philosophical question, they will not have an answer. Why do we even write $\exists x: \varphi(x)$, what is a function really? and so on. The questions and answers on this site are very representative of these notions. In the end, all these notions depend on certain beliefs. The very idea of logic, is that there is a process of reasoning which can be easily followed.
Dec
10
comment How to prove this logic?
The usual understanding is that mathematics is based on predicate logic. What logical system has unary relational symbols and allows for proof of predicate logic? This is not rigorous at all. I have given a proof where the system is well defined and each step is given within the system of predicate logic. Here there is no such system. And if it was there would have to be a mistake, because it is isomorphic to PL. Using concepts of higher algebra to prove concepts in PL is, confusing to say the least. How should relations be defined, if we use them to prove elementary theorems of PL...
Dec
10
comment What is a Complex Name?
The OP asked the question, because the problem is the definition and the exposition does not explain anything properly. What does it mean to apply predicate logic and what do the notations mean? Shouldn't a book on logic be clear, so that everyone with common sense should be able to follow it? The OP described the analysis of the expression as linguistics (which is actually logic). Perhaps everyone describing logic in this way, should call it something else. The m-logic: reasoning exclusively for mathematicians. Everyone else might want to learn logic.
Dec
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revised What is a Complex Name?
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Dec
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revised What is a Complex Name?
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Dec
10
answered What is a Complex Name?
Dec
10
comment What is a Complex Name?
This might help en.wikipedia.org/wiki/On_Denoting
Dec
7
comment How to prove this logic?
Note to reader. The accepted answer is meaningless and does not prove anything.