On Thu, 13 Jul 2006 11:18:22 -0400, Hatto von Aquitanien <abbot@AugiaDives.hre> 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. >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 >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.
Golly, it's a good thing you pointed that out! Most mathematicians don't realize this - for decades we've all been under the impression that anything that works for finite sets also works for infinite sets.
>>> 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? > >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.