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Show me the foundations for the continuum hypothesis. What you're arguing for is a view of mathematics that has been dead for a century now. With the Godels in

Show me the foundations for the continuum hypothesis. What you're arguing for is a view of mathematics that has been dead for a century now. With the Godels incompleteness theorem and Turing halting problem show you that there are cases of 'true' statements in mathematics that can't be reduced to "well-defined logical axioms".

ducttapecrown 8y ago
The continuum hypothesis is a well-defined logical axiom, occasionally.
Ivoirians 8y ago
My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions. Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. An aside for people who think Godel's incompleteness theorem somehow invalidates math: don't forget about the much lesser known Godel's completeness theorem, which states that if something is true in every model
Ivoirians 8y ago
My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions. Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. Godel's completeness theorem stipulates that there are self-contained consistent systems within the language wherein all of their nice proofs are true and universal, and where plenty of truth still needs to be discovered.
Ivoirians 8y ago
Re: Continuum hypothesis: the way I try to explain this is, take the statement S = "x^2 = -1 has no solutions". In Z, S is true. In C, S is false. What does it mean to say "S is true" without specifying the system? It depends on which underlying axioms you choose. There are extensions of ZFC where you can prove CH is true, and extensions where you can prove CH is false. In both extensions, the proofs are rigorous and valid, and there is plenty of valid mathematics to be done. But ok, my point is, how can you say the CH is a true statement that can't be reduced to axioms? What does it mean to be "true" besides that something follows from other truths? The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system.
- s-shellfish 8y ago
  There are two ways to see it. But that's also the problem. > What does it mean to be "true" besides that something follows from other truths? I agree with you, but I also don't. You know as well as I, that 'true' can mean something besides 'that which follows from other truths'. All the rules you have to rely on (such as'implication) - that's a truth you are dependent on for math to work, but can not define within math. Implication is a fundamental foundation. But can you define implication without using the concept of implication? > The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system. It's easy to point out flaws in reasoning. It's so, so much harder to have an airtight reasoning system, that goes for mathematics and computation, both, together, alone, etc. > What does it mean to be "true" besides that something follows from other truths? Truth is true, no more, no less. Once you turn it into symbols, it turns into a mess (or a work of art).

ducttapecrown 8y ago

The continuum hypothesis is a well-defined logical axiom, occasionally.

Ivoirians 8y ago

My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions. Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. An aside for people who think Godel's incompleteness theorem somehow invalidates math: don't forget about the much lesser known Godel's completeness theorem, which states that if something is true in every model

Ivoirians 8y ago

My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions. Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. Godel's completeness theorem stipulates that there are self-contained consistent systems within the language wherein all of their nice proofs are true and universal, and where plenty of truth still needs to be discovered.

Ivoirians 8y ago

Re: Continuum hypothesis: the way I try to explain this is, take the statement S = "x^2 = -1 has no solutions". In Z, S is true. In C, S is false. What does it mean to say "S is true" without specifying the system? It depends on which underlying axioms you choose. There are extensions of ZFC where you can prove CH is true, and extensions where you can prove CH is false. In both extensions, the proofs are rigorous and valid, and there is plenty of valid mathematics to be done. But ok, my point is, how can you say the CH is a true statement that can't be reduced to axioms? What does it mean to be "true" besides that something follows from other truths? The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system.

s-shellfish 8y ago
There are two ways to see it. But that's also the problem. > What does it mean to be "true" besides that something follows from other truths? I agree with you, but I also don't. You know as well as I, that 'true' can mean something besides 'that which follows from other truths'. All the rules you have to rely on (such as'implication) - that's a truth you are dependent on for math to work, but can not define within math. Implication is a fundamental foundation. But can you define implication without using the concept of implication? > The choice of axioms matters, sure, but everything that's true within a system follows from the axioms of that system. It's easy to point out flaws in reasoning. It's so, so much harder to have an airtight reasoning system, that goes for mathematics and computation, both, together, alone, etc. > What does it mean to be "true" besides that something follows from other truths? Truth is true, no more, no less. Once you turn it into symbols, it turns into a mess (or a work of art).