12 votes
Accepted

What are computable numbers, and what is their philosophical significance?

All mathematical formalizations of (intuitive) computability are known to be equivalent, in particular they are all equivalent to computability on the universal Turing machine. So technological ...
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  • 41k
11 votes
Accepted

Is there a way to avoid Gödel's incompleteness affecting mathematics as a whole?

It is a natural idea, but unfortunately the answer is no, it is not feasible. The root of incompleteness is not numbers, but the possibility of (implicit) self-reference, arithmetic is just the ...
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  • 41k
10 votes
Accepted

Are there rules for dealing with self-reference "paradoxes" in logic?

The same effect can be achieved with a single sentence:"This sentence is false". It is known as the Liar paradox and goes back to an ancient sophist Epimenides. Your two sentences simply split the ...
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  • 41k
8 votes
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What is the minimum number of axioms required for a system of axioms?

If we have an axiom system with a finite number of axioms, we can always reduce them to only one, replacing the set of original axioms with their conjunction. Thus, every non-trivial axiom system ...
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8 votes
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Where did Gödel write that first-order logic is the "true" logic?

He did not write it anywhere. The quote itself only calls Gödel and Skolem "alleged proponents", and later in the article Eklund remarks that "the (supposed) evidence that Skolem adhered to first-...
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8 votes
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What is the difference between logic and mathematics?

Whether there is a distinction, and what the distinction consists in, is a hotly debated topic. Here are a few things that are typically claimed to be essential to logic: Universal applicability: the ...
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  • 4,522
7 votes
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What are the "undefinable numbers" in real analysis and philosophy?

The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in ...
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  • 41k
6 votes
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Is there (or does something exist that is close to) a theory of arguments?

There is a theory of arguments, but I am afraid that the OP conception of argument is too idealized, and the notion of effective debate too narrow, to apply to most of them. If people argued from sets ...
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5 votes

Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?

There is no agreed upon example of this kind. Let us explore the issues: First, we cannot even come up with a decent example of a problem not solvable by any formal system. If you state the problem ...
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  • 764
5 votes
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Is it valid to prove the axioms of a system from themselves? How does it square with Gödel's incompleteness?

Yes, the axioms do trivially prove themselves. Your last derivation, however, is not valid: "A=A" can not be substituted for A because the latter is a symbol in a formal system, while the ...
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  • 41k
4 votes

Why is first-order logic defined as a collection of formal systems?

FOL is the "natural" logic environment to formalize mathematical theories. Propositional calculus, instead, is only a "toy": it is based on a very simplified "model" of language that is not useful to ...
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3 votes

What are computable numbers, and what is their philosophical significance?

The computable numbers are not technology dependent. A universal computer can simulate any finite physical system to any desired degree of accuracy. And it can simulate not just the input and the ...
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  • 7,053
3 votes

What are the "undefinable numbers" in real analysis and philosophy?

To paraphrase Joel Hamkins answer (pointed out by user4894) on the notion of undefinable reals. The naive account of undefinability points out there is only a countable number of ways we can describe ...
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3 votes

What is the minimum number of axioms required for a system of axioms?

GIT only applies to axiom systems that can express the natural numbers. Given any such one-axiom system, it would be incomplete. An example of a one-axiom system that's incomplete would be the ...
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  • 2,871
3 votes

What is(are) the importance(s) of formal reasoning

Formal reasoning is an important tool to make subtle differences transparent and explicit. This can be a prerequisite to assess the validity of certain argumentations. E.g., the correct definition of ...
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3 votes
Accepted

Was there a Kantian influence on Hilbert's formalist programme?

I think the confusion comes from schematic identification of logicism with realism, intuitionism with conceptualism, and formalism with nominalism, referencing positions in the old debate on the ...
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3 votes
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Prove a logical formula is equivalent to the contradiction if and only if the set it describes is empty

We have to remember that : a ∈ {x:ψ(x)} ⇔ ψ(a) (this is the definition of the "set-builder" symbol { _ : __ } ). But {x:ψ(x)} = Ø, and thus : for all a, a ∉ {x:ψ(x)} ⇔ for all a, ¬ψ(a).
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3 votes

What are the "undefinable numbers" in real analysis and philosophy?

Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "...
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3 votes

How does abstraction/generalization in mathematics fit into inductive reasoning?

You're correct that moving from the integers to the rationals does not fit, because the generalisation that inductive reasoning refers to is a generalisation of statements (or predicates) not of the ...
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  • 506
3 votes

Logic as a (bad) model language

It is reasonable to mention Wittgenstein in this connection. Brouwer is also relevant as he saw logic as secondary to mathematics. In Brouwers case it is rather clear-cut that he rejected logic as a ...
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3 votes

Formal proof : predicate logic

I would do the following: 1. ∃x Cube(x) ∧ Small(d) 2. ∃x Cube(x) ∧Elim1 3. Small(d) ∧Elim1 4. | Cube(z) A 5. | Cube(z) ^ Small(d) ^Intr 3,4 6. | ∃x(Cube(x) ^ Small(d)) ∃Intr 5 ...
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3 votes

Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?

Every formal system was once the creative construction of an individual or a group of individuals to solve a problem that an existing formal system had not already solved. The best example I can give ...
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3 votes
Accepted

Formal proof errors

Edit You can only use conjunctive, disjunctive, and negation intro/elim and only uses TautCon for DeMorgans. You also seem to be using rules called "contradiction introduction", and "contradiction ...
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  • 2,311
3 votes

Why is there no "Mario Bunge" entry in the SEP? Where could I find a valuable presentation of Mario Bunge's philosophy?

There are two questions here (rephrased): "Where can I find an accessible introduction to Bunge's philosophy?" and "Why is Bunge not more frequently referred to?" Let's start with ...
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  • 151
3 votes

Does try-catch from programming has any basis in formal logic or mathematics and if so what is it?

There are many connections between programming and formal logic. The Curry-Howard correspondence gives a direct relationship between computer programs and formal proofs. Exception-handling, ...
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  • 7,025
2 votes
Accepted

Logic as a (bad) model language

Hilary Putnam argues that formal systems cannot even be used to perform hypothesis selection. Whether you wish to characterize this as "prescription" is up to you, but one might suspect that if formal ...
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  • 2,961
2 votes

Logic as a (bad) model language

For your question I would recommend the preface of Henri Poincaré's Science and Hypothesis (1905), particularly this piece: For [in mathematics] the mind may affirm because it lays down its own ...
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  • 821
2 votes

Is there a formal presentation of Maturana's "Autopoiesis"

There exists a rather obscure line of mathematical Biology called "Relational Biology" which connects the names Nicolas Rashevsky, Richard Rosen and Aloisius Louie, involving a relational, algebraic ...
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