On Jun 15, 6:04 pm, Charlie-Boo <shymath...@gmail.com> wrote: > On Jun 15, 7:31 am, Zuhair <zaljo...@gmail.com> wrote: > > > #1 Don't try to define math in terms of math. Besides being circular > reasoning, you are simply taking a subset of math, isolating and > formalizing it. Define math in informal non-mathematical terms. > > # 2. Know what level of abstraction you are at. Don't define science, > or communicating. Talking about strings of symbols being generated > above sounds like a model of communicating, the wrong level of > abstraction. >
No a model of communicating is string of symbols 'expressing' non symbolic stuff usually, while here I'm speaking about a machinery of producing symbols from symbols, this is not about communication.
> # 3. And this is especially for you Zuhair: State exactly the goal, > and then the approach before you are formal at all. Getting into > formalizing without a clear idea of how you are going to approach/ > solve your problem/goal is like programmers who write software without > specifications (never mind technical design) or programming language > committees who require all proposals to be in the form of actual > constructs and syntax, rather than goals, approaches, and then > formalization. Or like talking about (YOU) formalizing a resolution > to the liar paradox without having a clear idea (or any idea) how you > intend to resolve it. You generate solutions after you build an > axiomatic system, not before. > > C-B
What has been referred to in the headpost is the question of whether second order arithmetic can be interpreted in a system that is most naively understood as a LOGICAL system. The system I spoke about naturally and naively extends known logical systems, it is only deriven by setting rules of formation of formulas of the language as to avoid paradoxes and without any unnecessary concepts outside of logic, so it would be a logical system. You may say that this system is a sector of mathematics that I call as logic and that I'm reducing the other sectors of mathematics to that sector, OK if you understand it that way, no problem, the issue is whether this can be done, and the above record showes that it can be done and naively so.
However I consider *all* parts of first order logic as being LOGICAL and not mathematical, yes it uses symbols and ways commonly used in mathematics like function symbols, variables, constants, natural indices, recursive machinery, etc... *ALL* of those I maintain as being legitamate parts of LOGIC, i.e. they are LOGICAL symbols and machineries, that mathematics also uses those is just an overlap, it gives no priority to mathematics over logic.