We might want to start a new thread, I don't want to hijack Messiah's thread. Either math or logic is fine!
Yes, I agree. Actually I think I've really sadi all I can say on the subject. Any more and I'd just repeating myself. I am obviously no expert but am willing to continue on another thread if you think it worthwile.
Take a break and listen to this lecture by Sir Penrose, which includes his views on philosophy of mathematics and natural sciences. Specifically, at slides 09 and 10 is something called Goodstein's theorem. It defies standard induction technique but yields nicely to general ordinal arithmetic.
That very interesting where Penrose suggests that thoughts and free will may stem from principles in physics, but it's seems very fringe science. I don't think I would have a clue what they were suggesting without the diagrams.
Does logic require a concrete defintion of a thing to justify that it can be deduced logically? Is there ever a perfect defenition of anything?
Logic is slow like science but accurate, in my view that is the main impeding limit of logic, there is likely much more that we have a notion of but haven't worked out logically or mathematically yet because many things are so abstract and some causes and effects separated by months or years that we sense them in other ways long before they are awaringly reasoned out.
Logic is a derivative of reality.
Some short observations about logic.In fact logic is the branch of philosophy which deals with correct reasoning that's it with the correct deduction of conclusions from a given set of premises.Observe that there is no need for the premises to be true empirically.Basically logic is a feature of human reasoning,very useful in epistemology indeed,but there is no necessity that nature should obey the rules of logic.A good example is quantum physics,more specifically the violation of Bell's inequalities problem,where there is still valid an option often overlooked by a majority of scientists (for pragmatic reasons):that usual logic could be invalid when applied to explain phenomena at that level (in the sense that it cannot give us a good description of reality since the nature does not follow its rules).
I do not know whether you are accustomed with the philosophy of religion but one of the main objections at all deductive arguments pro/con God hypothesis is exactly that:even if we had a sound,irrefutable logically,argument pro/con God there is no necessity to believe/disbelieve without having empirical arguments beyond all reasonable doubt proving that God does exist/not exist.So that even if an argument pro God were sound that does not imply that God does exist in reality (valid also for the arguments against God's existence).Sounds strange but even if the premises of very succesfull scientific theories are all true (empirically) there is no necessity that all their conclusions are also true empirically.That's why for example it is still reasonable to doubt all predictions made by a very succesfull otherwise scientific theories not 'confirmed' practically with arguments beyond all reasonable doubt.For example General Relativity predicts that in the singularity of a black hole it ceases to give an accurate description of empirical facts,still we do not have yet sufficient arguments which to compel us to believe that,so that skepticism about this is still rational.