Re: Does the Calculus rest on Euclid?

Gene Ward Smith wrote:


Virgil wrote:

I note that Dededkind and Cauchy, whose names were invoked in this
discussion, preceded Weyl.

Weyl died in 1955. Presumably the work cited above was published before
that so that there has been over half a century of progress since.

Weyl was one of those philosophical gloomy gusses who worried about
impredicative definitions. Then he got into Intuitionism. These days
people of that sort are likely to embrace Constructivism. Most
mathematicians think mathematics is hard enough without artificially
making it harder because of unconvincing philsophical qualms, but
pissing and moaning like Hatto does has a long tradition behind it. Why
types don't solve his problem I don't know.

Perhaps types _do_ solve this problem, but that solution has not been
explained to me. But as far as pessimism goes, I don't believe it is
applicable to my view on all of this. I believe that all of mathematics
rests upon some very basic orthogonal propositions a priori. These include:

An assertion that is not ture is false.
The concept of cutability.
The concept of countability.
The concept of completed infinity.

Though my set of judgments a-priori may differ from Kant's, I believe a
correct understanding of Kant will show that he was correct regarding the
idea that the mind relies on certain basic ideas which exist a priori.
--
Nil conscire sibi
.

Relevant Pages

  • Re: Does the Calculus rest on Euclid?
    ... |>> discussion, preceded Weyl. ... "gloomy gus" seems to be the generic way to try to make someone ... Intuitionism is a kind of constructivism. ... swathes of mathematics impossible to do. ...
    (sci.math)
  • Re: Attempts to Refute Cantors Uncountability Proof?
    ... If I could lay my hands on a copy of Weyl's _Philosophy of Mathematics and ... Weyl said seems clear enough to convince me that his objections are ... The "statement" is, so far as I can tell, not a quote from Weyl. ... the quote doesn't occur in the same context and I can't locate it. ...
    (sci.math)
  • Re: Does a potential infinity actually exist?
    ... Hermann Weyl: "... ... mathematics, and finally applied it, without justification, to the ... So the basic idea, as so eloquently stated by Weyl, is that infinite ... finite sets exist, and that mathematics should distinguish between the ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... Weyl died in 1955. ... I believe that all of mathematics ... correct understanding of Kant will show that he was correct regarding the ... It follows that anyone relying on a priori 'truths' runs some risk of ...
    (sci.math)