Hatto von Aquitanien wrote: > guenther vonKnakspot wrote: > > > Hatto von Aquitanien wrote: > >This > > is not howewer the flawed reasoning that I am refering to. I am talking > > much baser contentions made by the ill educated like denying the > > existence of certain mathematical objects on the ground that they can > > not be physically constructed. An example would be the set of Natural > > Numbers whichs existence is denied because there are not sufficient > > atoms in the universe to build a tangible physical representation of > > it, or specific irrational numbers on on the same grounds pertaining to > > their decimal base representation. > > I don't know that such an opinion is ill-educated or merely unpersuasive. It is not a matter of what the opinion itself is. The problem is with the person holding such an opinion. That person must have received a flawed or no mathematical education at all.
> OTOH, I don't believe that is the essence of most objections to the concept > of infinite sets. For myself, I am willing to consider the existence of Of course it is not the essence of most objections to to the concept of infinite sets. It is an example of the flawed assumption that mathematics are constrained by physical reality in such a manner.
> infinite sets without much reservation. At the same time, I believe that > arguments and reasoning applicable to finite sets should be applied with > caution to infinite sets, and carefully scrutinized to determine if it > makes sense in any given instance. That is not related to this discussion.
> >> that mathematics should not be required to produce physically measurable > >> results as a test of its validity. I really have to wonder if such a > >> requirement is unrealistic. > > > > If you make such a requirement, then you will not get very far. What is > > a physically measurable result that gives validity to the number 2 ? Or > > to the law of distributivity? or to the differentiability of a given > > function? Mathematical concepts have no consequences in physical > > reality and the laws of the physical universe have no consequence for > > mathematical concepts. (as long as we keep to the agreement I requested > > above). > > I am not stating this conclusively, but consider Cantor's transfinite > induction, in comparison to the calculus of Newton and Leibnitz. We have > ample evidence that the results of differentiation and integration > correspond in many cases to physical experiments in convincing ways. Is > there any result from Cantor's theory which leads to a verifiable > prediction in terms of physical experiment? You were the one arguing in favour of the requirement that mathematical concepts be subjected to verification by physical measurability. Obviously you have found a further difficulty in supporting that case.
> That is merely one aspect of the idea that mathematics might be physically > testable. Another is whether the actual reasoning can be reproduced or > verified using computers. > > >> It's interesting to observe that some people > >> are wont to point to the fact that formal proofs can be verified by > >> computer programs. > >> > >> I note that you object to the idea that "mathematics is dependent on > >> computability". I don't know if you mean that exclusively in terms of > >> solving equations and/or finding approximate numerical solutions, or if > >> you also object to the idea the mathematical proofs should be machine > >> verifiable. I curious to know what you think of this: > >> http://metamath.org/ > > > > I agree with the idea that mathematical proofs should be machine > > verifiable; as long as this is a statement about computer science and > > not about mathematics. > > One might argue that this is merely using the computer as a tool. Your use of the word 'merely' suggests that you might be thinking of manners of using a computer trascending those of a tool. What would those be ?