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It seems to me that the absolute intuitionist and absolute logician can not exist as real persons. I see them rather as two extremes that are always mixed in so

by dnc 13y ago

It seems to me that the absolute intuitionist and absolute logician can not exist as real persons. I see them rather as two extremes that are always mixed in some proportion (and always in fight with each other for more space, so the proportion changes over time). For instance, Euclid is mentioned as logician, but at least one of his axioms (about parrallel lines that could not intersect) is based on intuition that, turns out, is not always "true". Therefore we have got other Non-Euclidian geometries.

thetwiceler 13y ago
Actually, this example makes Euclid even more of a logician. Recall that Euclid stated his Fifth Postulate (where his postulates really mean axioms). Geometers for thousands of years afterward, guided by intuition, attempted to prove that in fact the Fifth Postulate could be proved from the other axioms. But by the 19th century, it was actually clear that Euclid was correct to include the postulate; had he not included the postulate, his geometry would not convey what he wanted to convey. Euclid in this way was fundamentally a logician! He realized that he needed the Fifth Postulate to prove facts about a geometry he imagine (e.g. angles in a triangle sum to two right angles), and included as an axiom something that others intuited must be derivable from the others. But as Poincare mentions, we can also see that Euclid DID have much "intuition" in his works. In modern days, we do not consider Euclid's Elements a rigorous work of logic, mainly because his definitions are not rigorous definitions; he says a point is "that which has no part." Hilbert remedied Euclidean geometry in the 20th century, with a work that has some undefined objects (points and lines) and precisely defines the rest. He needs something like 24 axioms, as opposed to Euclid's 5. It's interesting to see where Euclid's logic breaks down. Look at his very first proof - the construction of an equilateral triangle. He constructs two circles and looks at their intersection. How do we know we can build the circles to intersect? He draws us a picture and it seems obvious :). But in Hilbert's Euclidean geometry, we need what I think of as a really nasty axiom in order to ensure the circles intersect: "Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid."
- dnc 13y ago
  Thanks for pointing out this.

by dnc 13y ago

thetwiceler 13y ago

Actually, this example makes Euclid even more of a logician. Recall that Euclid stated his Fifth Postulate (where his postulates really mean axioms). Geometers for thousands of years afterward, guided by intuition, attempted to prove that in fact the Fifth Postulate could be proved from the other axioms. But by the 19th century, it was actually clear that Euclid was correct to include the postulate; had he not included the postulate, his geometry would not convey what he wanted to convey. Euclid in this way was fundamentally a logician! He realized that he needed the Fifth Postulate to prove facts about a geometry he imagine (e.g. angles in a triangle sum to two right angles), and included as an axiom something that others intuited must be derivable from the others. But as Poincare mentions, we can also see that Euclid DID have much "intuition" in his works. In modern days, we do not consider Euclid's Elements a rigorous work of logic, mainly because his definitions are not rigorous definitions; he says a point is "that which has no part." Hilbert remedied Euclidean geometry in the 20th century, with a work that has some undefined objects (points and lines) and precisely defines the rest. He needs something like 24 axioms, as opposed to Euclid's 5. It's interesting to see where Euclid's logic breaks down. Look at his very first proof - the construction of an equilateral triangle. He constructs two circles and looks at their intersection. How do we know we can build the circles to intersect? He draws us a picture and it seems obvious :). But in Hilbert's Euclidean geometry, we need what I think of as a really nasty axiom in order to ensure the circles intersect: "Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid."

dnc 13y ago
Thanks for pointing out this.