[Agda] If all functions (N->N)->N are continuous, then 0=1.
Altenkirch Thorsten
psztxa at exmail.nottingham.ac.uk
Thu Nov 28 21:50:44 CET 2013
I would put it rather positively: Type Theory enables us to have two
different forms of existence: the proof-relevant one which corresponds to
the statement "there exists" and which isn't necessarily a proposition and
the proof-irrelevant one which (following Mile Shulmann) we can paraphrase
as "there merely exists".
Hence we know that type-theoretically the correct way to express
continuity is using "merely exists" but not "exists". Actually a similar
situation arises which Church thesis which should say for every function
there merely exists a code which realizes it.
One is left wondering wether there is a better way expressing these
principles since propositional principles feel a bit alien in type theory.
Thorsten
On 28/11/2013 09:57, "Thierry Coquand" <Thierry.Coquand at cse.gu.se> wrote:
>
> It is tempting to have the following argument
>
> -when Brouwer formulated the continuity axiom, expressed in Martin's
>message as
>"the value f(a) can depend only on a finite prefix of the infinite
>argument a" which can be reformulated as "there exists n such that if a
>k = b k
>for all k<n then f(a) = f(b)" this existence statement was understood
>intuitionistically
>
> -hence Martin's message shows that one cannot express
>constructive/intuitionistic
>existence using sigma types
>
> Thierry
>
>On Nov 28, 2013, at 9:28 AM, Altenkirch Thorsten wrote:
>
>> I am puzzled again!
>>
>> Clearly, this formulation of continuity is inconsistent with functional
>> extensionality because it makes it possible to observe an intensional
>> property of a function - its modulus of continuity. What is surprising
>>is
>> that you manage to derive a contradiction w.o. using ext! So the wrong
>> formulation of continuity doesn't even work in vanilla intensional Type
>> Theory.
>>
>> Thorsten
>>
>> On 27/11/2013 21:56, "Martin Escardo" <m.escardo at cs.bham.ac.uk> wrote:
>>
>>>
>>>
>>>
>>> If all functions (N->N)->N are continuous, then 0=1.
>>
>> That is a bit of a lie. What you are saying is if for all functions we
>> know why they are continuous, then 0=1.
>>
>> I mean "is continuous" does sound like a proposition, doesn't it?
>>
>>>
>>>
>>>
>>> Think of functions a : N -> N as sequences of natural numbers:
>>>
>>> a 0, a 1, a 2, a 3, ..., a n, ...
>>>
>>> The "continuity axiom" for functions
>>>
>>> f : (N -> N) -> N,
>>>
>>> that map sequences a : N -> N to numbers f(a), going back to Brouwer
>>> in his intuitionistic mathematics in the early 20th century, says that
>>> the value f(a) can depend only on a finite prefix of the infinite
>>> argument a.
>>>
>>> This makes sense computationally: there are no crystal balls in
>>> computational processes, able to see and grasp the infinite input at
>>> once. The input of f is computed bit-by-bit, in fact, often only when
>>> f queries it.
>>>
>>> After finitely many queries to its argument a : N -> N, the function f
>>> is supposed to deliver its answer, if it's ever going to produce an
>>> answer.
>>>
>>> When this finiteness condition holds, we say that f is continuous
>>> (never mind the reason for the terminology "continuous" in this
>>>message).
>>>
>>> The link
>>>
>>> http://www.cs.bham.ac.uk/~mhe/continuity-false/continuity-false.html
>>>
>>> writes down Brouwer's formulation of continuity in Agda notation, and
>>> proves that:
>>>
>>> If all functions (N->N)->N are continuous, then 0=1.
>>>
>>> What is going on? There is some discussion in the above link.
>>>
>>> You may wish to read Andrej Bauer's related discussion
>>>
>>>http://math.andrej.com/2011/07/27/definability-and-extensionality-of-the
>>>-m
>>> odulus-of-continuity-functional/
>>>
>>> The ideas of http://homotopytypetheory.org/book/ are relevant in this
>>> respect too. To solve the apparent paradox, think about how one can
>>> constructively prove existence without disclosing witnesses. Doubly
>>> negated existence is too weak. One needs a stronger, constructive
>>> notion of existence that deliberately doesn't exhibit witness, which
>>> sometimes, but not always, can get to be disclosed. I attended a nice
>>> talk by Thierry Coquand yesterday, in which he showed how the "axiom
>>> of description" explains when and how the secret can be constructively
>>> disclosed. In the HoTT book, this is explained as the axiom of unique
>>> choice.
>>>
>>> NB. Only the purest form of intensional Martin-Loef Type Theory is
>>> used in the above Agda proof, and no universes are needed. No HoTT
>>> axioms in particular.
>>>
>>> Best, Martin
>>>
>>> _______________________________________________
>>> Agda mailing list
>>> Agda at lists.chalmers.se
>>> https://lists.chalmers.se/mailman/listinfo/agda
>>
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