I’ve rarely seen disagreement on basic axioms. p → p seems to be rather uncontroversial, although it’s based on ‘intuition’. On the other hand, that’s the purpose of deduction: reduce the need of intuition only to the smallest and least controversial set of assertions. This does not imply, as your original formulation seems to, that then intuition can be used for everything.
his does not imply, as your original formulation seems to, that then intuition can be used for everything.
I never said anything of the kind. My point was that intuition is unavoidably involved in everything.
I’ve rarely seen disagreement on basic axioms.
Then check out the controversy over Euclid’s fifth postulate, mathematical intuitionism, the Axiom of Choice, whether existence is a predicate, etc, etc.
. p → p seems to be rather uncontroversial, although it’s based on ‘intuition’.
Some would say that it’s based on truth tables,and defies intuition!
See the logical versus material implication controversy:-
I never said anything of the kind. My point was that intuition is unavoidably involved in everything.
On this I think we agree. I’ll just add that sometimes “intuitions” points to “a short mental calculation” and some other times to “a biased heuristic”. The fact that we don’t have access to which is which is the danger of accepting sentences like “intuition is the basis of everything”. I would rather prefer two different words for the two different kind of intuitions, but there aren’t.
Then check out the controversy over Euclid’s fifth postulate, mathematical intuitionism [...]
Yes, but there are also never been controversy on the first postulate… Some axioms are more basic than others. And indeed challenged axioms produce strong revolutions.
Some would say that it’s based on truth tables,and defies intuition!
This I don’t know how to interpret. Truth table are useful as long as they agree on the axioms. Or one could say that truth tables are based on intuition...
On this I think we agree. I’ll just add that sometimes “intuitions” points to “a short mental calculation” and some other times to “a biased heuristic”.
It can mean either of those, but it can also mean an assumption you can neither prove nor do without.;
Some axioms are more basic than others.
If what you want is convergence on objective truth, it is the existence of axioms that people don’t agree on that is the problem.
And indeed challenged axioms produce strong revolutions.
And pluralism. Intuitonistic and classical maths co-existing, Euclidean and non-Euclidean geometry co-exisitng.
. Truth table are useful as long as they agree on the axioms. Or one could say that truth tables are based on intuition...
Truth tables give you a set of logical functions, some of which resemble traditional logical connectives, such as “and” and “implies” to some extent. But only to some extent. The worry is that they don’t capture all the features of ordinary langauge usage.
I’ve rarely seen disagreement on basic axioms. p → p seems to be rather uncontroversial, although it’s based on ‘intuition’.
On the other hand, that’s the purpose of deduction: reduce the need of intuition only to the smallest and least controversial set of assertions. This does not imply, as your original formulation seems to, that then intuition can be used for everything.
I never said anything of the kind. My point was that intuition is unavoidably involved in everything.
Then check out the controversy over Euclid’s fifth postulate, mathematical intuitionism, the Axiom of Choice, whether existence is a predicate, etc, etc.
Some would say that it’s based on truth tables,and defies intuition!
See the logical versus material implication controversy:-
http://www.askphilosophers.org/question/4103
On this I think we agree. I’ll just add that sometimes “intuitions” points to “a short mental calculation” and some other times to “a biased heuristic”. The fact that we don’t have access to which is which is the danger of accepting sentences like “intuition is the basis of everything”.
I would rather prefer two different words for the two different kind of intuitions, but there aren’t.
Yes, but there are also never been controversy on the first postulate… Some axioms are more basic than others. And indeed challenged axioms produce strong revolutions.
This I don’t know how to interpret. Truth table are useful as long as they agree on the axioms. Or one could say that truth tables are based on intuition...
It can mean either of those, but it can also mean an assumption you can neither prove nor do without.;
If what you want is convergence on objective truth, it is the existence of axioms that people don’t agree on that is the problem.
And pluralism. Intuitonistic and classical maths co-existing, Euclidean and non-Euclidean geometry co-exisitng.
Truth tables give you a set of logical functions, some of which resemble traditional logical connectives, such as “and” and “implies” to some extent. But only to some extent. The worry is that they don’t capture all the features of ordinary langauge usage.