[FOM] "Hidden" contradictions
henk at cs.ru.nl
Mon Aug 19 14:06:47 EDT 2013
The inconsistency is having infinitesimals
h <1/n, for all n,
but with h=/=0. This is impossible for a real number.
The later added infinitesimals provide one way out.
To Antonino Drago:
All right: Cauchy did the first step and Weiserstrass and Dedkind
Poincare' calls Weiserstrass a 'logician', in a somewhat negative sense
(meaning less creative than mathematicians); but then he adds: 'but we
need logicians'. I'd agree with both statements.
On 08/17/2013 10:39 PM, Antonino Drago wrote:
> Cauchy did not removed the inconsistency, because the more basic
> theory of the real numbers came fifty years later, through the works
> by Weierstrass and Dedekind.
> Hence the period of progressive advancement of mathematics
> notwithstanding of the inconsistencies was even more longer.
> Antonino Drago
> ----- Original Message ----- From: "henk" <henk at cs.ru.nl>
> To: <fom at cs.nyu.edu>
> Sent: Friday, August 16, 2013 12:18 AM
> Subject: Re: [FOM] "Hidden" contradictions
>> Calculus by Leibniz is inconsistent. Yet using this theory he, and
>> notably Euler, made---with the right intuition---wonderful
>> predictions. It took Cauchy (epsilon delta) and later non-standard
>> analysis to remove the (most obvious) inconsistency.
>> The use of this later work is to make it more easy to formulate what
>> is the right intuition.
>> On 08/15/2013 04:35 AM, Timothy Y. Chow wrote:
>>> On Wed, 14 Aug 2013, Mark Steiner wrote:
>>>> I appreciate this response. However, my physicist friends tell me
>>>> that the theory known as QED is thought to be inconsistent, but
>>>> people use it anyway, with great success in predictions. I think
>>>> what this means is the claim that there is no way to formalize QED
>>>> in a consistent axiomatic system. If this is right, then there is
>>>> a sense in which formal systems do play some kind of role in physics.
>>> The alleged inconsistency of QED is a complicated topic that has
>>> been discussed in great detail before on FOM and I don't think we
>>> want to rehash it all here, but I'll just say that even if we grant
>>> the (somewhat debatable) propositions that (1) "QED is thought to be
>>> inconsistent" and (2) this means that "there is no way to formalize
>>> QED in a consistent axiomatic system", then really all this shows is
>>> the exact opposite: namely, that formal systems *do not* play an
>>> important role in physics. If they did, then the physicists would be
>>> compelled to abandon QED. The only role formal systems are playing
>>> here is in framing certain philosophical discussions *about* physics.
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