Bishop's work on type theory

Bishop's work on type theory

[Martín Hötzel Escardó]

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This week I learned two interesting things that seem to be kept as a 
guarded secret:

(1) Errett Bishop reinvented type theory.
(2) He also explained how to compile it to Algol.

I am adding a link to these two manuscripts. A nice quote from the second 
paper (Algol.pdf) is this, in my opinion, because it foresees things such 
as Agda, Coq, NuPrl, ...

"The possibility of such a compilation demonstrates the existence of a new 
type of programming language, one that contains theorems, proofs, 
quantifications, and implications, in addition to the more conventional 
facilities for specifying algorithms"

This was in the late 1960's (or correct me). Here is a link to both 
manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/

Greetings from Bonn.
Martin


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Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

Very interesting!!

I believe it's been said that Martin-Lof developed his type theory
explicitly with the goal of providing a foundation for Bishop's
constructive mathematics.  Was he aware of these manuscripts of
Bishop's at the time?

Also: does this conclusively prove that Bishop's "sets" coincide with
what type theorists generally call "setoids"?


On 5/4/18, Martín Hötzel Escardó <escardo...@gmail.com> wrote:
> This week I learned two interesting things that seem to be kept as a
> guarded secret:
>
> (1) Errett Bishop reinvented type theory.
> (2) He also explained how to compile it to Algol.
>
> I am adding a link to these two manuscripts. A nice quote from the second
> paper (Algol.pdf) is this, in my opinion, because it foresees things such
> as Agda, Coq, NuPrl, ...
>
> "The possibility of such a compilation demonstrates the existence of a new
> type of programming language, one that contains theorems, proofs,
> quantifications, and implications, in addition to the more conventional
> facilities for specifying algorithms"
>
> This was in the late 1960's (or correct me). Here is a link to both
> manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>
> Greetings from Bonn.
> Martin
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.
>

Re: [HoTT] Bishop's work on type theory

[Bas Spitters]

Hi Martin,

These were discussed publically at some point. I've got them at around 2000.
We never put them on the web, because Bishop had decided not to publish them.
Since you are doing this now, it might be good to at least add a note
to that respect, so that people can put them in context.

See you in Bonn!

Bas

On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó
<escardo...@gmail.com> wrote:
> This week I learned two interesting things that seem to be kept as a guarded
> secret:
>
> (1) Errett Bishop reinvented type theory.
> (2) He also explained how to compile it to Algol.
>
> I am adding a link to these two manuscripts. A nice quote from the second
> paper (Algol.pdf) is this, in my opinion, because it foresees things such as
> Agda, Coq, NuPrl, ...
>
> "The possibility of such a compilation demonstrates the existence of a new
> type of programming language, one that contains theorems, proofs,
> quantifications, and implications, in addition to the more conventional
> facilities for specifying algorithms"
>
> This was in the late 1960's (or correct me). Here is a link to both
> manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>
> Greetings from Bonn.
> Martin
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.

Re: [HoTT] Bishop's work on type theory

[Martín Hötzel Escardó]

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Hi Bas,

Perhaps, to have this in context, we could add it to e.g. the HoTT web page 
and/or the nlab. 

Do you know precise dates for these manuscripts?

I am looking forward to seeing you in Bonn.

Also, it would be nice to have Mike Shulman's questions answered or at 
least addressed.

Martin

On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote:
>
> Hi Martin, 
>
> These were discussed publically at some point. I've got them at around 
> 2000. 
> We never put them on the web, because Bishop had decided not to publish 
> them. 
> Since you are doing this now, it might be good to at least add a note 
> to that respect, so that people can put them in context. 
>
> See you in Bonn! 
>
> Bas 
>
> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó 
> <escar...@gmail.com <javascript:>> wrote: 
> > This week I learned two interesting things that seem to be kept as a 
> guarded 
> > secret: 
> > 
> > (1) Errett Bishop reinvented type theory. 
> > (2) He also explained how to compile it to Algol. 
> > 
> > I am adding a link to these two manuscripts. A nice quote from the 
> second 
> > paper (Algol.pdf) is this, in my opinion, because it foresees things 
> such as 
> > Agda, Coq, NuPrl, ... 
> > 
> > "The possibility of such a compilation demonstrates the existence of a 
> new 
> > type of programming language, one that contains theorems, proofs, 
> > quantifications, and implications, in addition to the more conventional 
> > facilities for specifying algorithms" 
> > 
> > This was in the late 1960's (or correct me). Here is a link to both 
> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/ 
> > 
> > Greetings from Bonn. 
> > Martin 
>
>

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Re: [HoTT] Bishop's work on type theory

[Bas Spitters]

Setoids were introduced by Martin Hofmann is his PhD-thesis. They were
"inspired" by Bishop; see p8:
www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/ECS-LFCS-95-327.ps

On Sat, May 5, 2018 at 12:04 AM, Martín Hötzel Escardó
<escardo...@gmail.com> wrote:
> Hi Bas,
>
> Perhaps, to have this in context, we could add it to e.g. the HoTT web page
> and/or the nlab.
>
> Do you know precise dates for these manuscripts?
>
> I am looking forward to seeing you in Bonn.
>
> Also, it would be nice to have Mike Shulman's questions answered or at least
> addressed.
>
> Martin
>
> On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote:
>>
>> Hi Martin,
>>
>> These were discussed publically at some point. I've got them at around
>> 2000.
>> We never put them on the web, because Bishop had decided not to publish
>> them.
>> Since you are doing this now, it might be good to at least add a note
>> to that respect, so that people can put them in context.
>>
>> See you in Bonn!
>>
>> Bas
>>
>> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó
>> <escar...@gmail.com> wrote:
>> > This week I learned two interesting things that seem to be kept as a
>> > guarded
>> > secret:
>> >
>> > (1) Errett Bishop reinvented type theory.
>> > (2) He also explained how to compile it to Algol.
>> >
>> > I am adding a link to these two manuscripts. A nice quote from the
>> > second
>> > paper (Algol.pdf) is this, in my opinion, because it foresees things
>> > such as
>> > Agda, Coq, NuPrl, ...
>> >
>> > "The possibility of such a compilation demonstrates the existence of a
>> > new
>> > type of programming language, one that contains theorems, proofs,
>> > quantifications, and implications, in addition to the more conventional
>> > facilities for specifying algorithms"
>> >
>> > This was in the late 1960's (or correct me). Here is a link to both
>> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>> >
>> > Greetings from Bonn.
>> > Martin
>>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.

Re: [HoTT] Bishop's work on type theory

[Martín Hötzel Escardó]

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(I know that, and probably Mike knows that too. Martin)

On Saturday, 5 May 2018 00:12:51 UTC+2, Bas Spitters wrote:
>
> Setoids were introduced by Martin Hofmann is his PhD-thesis. They were 
> "inspired" by Bishop; see p8: 
> www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/ECS-LFCS-95-327.ps 
>
> On Sat, May 5, 2018 at 12:04 AM, Martín Hötzel Escardó 
> <escar...@gmail.com <javascript:>> wrote: 
> > Hi Bas, 
> > 
> > Perhaps, to have this in context, we could add it to e.g. the HoTT web 
> page 
> > and/or the nlab. 
> > 
> > Do you know precise dates for these manuscripts? 
> > 
> > I am looking forward to seeing you in Bonn. 
> > 
> > Also, it would be nice to have Mike Shulman's questions answered or at 
> least 
> > addressed. 
> > 
> > Martin 
> > 
> > On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote: 
> >> 
> >> Hi Martin, 
> >> 
> >> These were discussed publically at some point. I've got them at around 
> >> 2000. 
> >> We never put them on the web, because Bishop had decided not to publish 
> >> them. 
> >> Since you are doing this now, it might be good to at least add a note 
> >> to that respect, so that people can put them in context. 
> >> 
> >> See you in Bonn! 
> >> 
> >> Bas 
> >> 
> >> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó 
> >> <escar...@gmail.com> wrote: 
> >> > This week I learned two interesting things that seem to be kept as a 
> >> > guarded 
> >> > secret: 
> >> > 
> >> > (1) Errett Bishop reinvented type theory. 
> >> > (2) He also explained how to compile it to Algol. 
> >> > 
> >> > I am adding a link to these two manuscripts. A nice quote from the 
> >> > second 
> >> > paper (Algol.pdf) is this, in my opinion, because it foresees things 
> >> > such as 
> >> > Agda, Coq, NuPrl, ... 
> >> > 
> >> > "The possibility of such a compilation demonstrates the existence of 
> a 
> >> > new 
> >> > type of programming language, one that contains theorems, proofs, 
> >> > quantifications, and implications, in addition to the more 
> conventional 
> >> > facilities for specifying algorithms" 
> >> > 
> >> > This was in the late 1960's (or correct me). Here is a link to both 
> >> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/ 
> >> > 
> >> > Greetings from Bonn. 
> >> > Martin 
> >> 
> > -- 
> > You received this message because you are subscribed to the Google 
> Groups 
> > "Homotopy Type Theory" group. 
> > To unsubscribe from this group and stop receiving emails from it, send 
> an 
> > email to HomotopyTypeThe...@googlegroups.com <javascript:>. 
> > For more options, visit https://groups.google.com/d/optout. 
>

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Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

Right, the question more precisely is whether, when transported along
whatever isomorphism there is between Bishop's "general language" and
MLTT (I have not read the manuscript yet to understand this), the
"sets" defined by Bishop on p16 coincide with Hofmann's setoids.  If
so, then it would be some substantial additional evidence for the
claim that setoids are "what Bishop really meant".

On 5/4/18, Martín Hötzel Escardó <escardo...@gmail.com> wrote:
> (I know that, and probably Mike knows that too. Martin)
>
> On Saturday, 5 May 2018 00:12:51 UTC+2, Bas Spitters wrote:
>>
>> Setoids were introduced by Martin Hofmann is his PhD-thesis. They were
>> "inspired" by Bishop; see p8:
>> www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/ECS-LFCS-95-327.ps
>>
>> On Sat, May 5, 2018 at 12:04 AM, Martín Hötzel Escardó
>> <escar...@gmail.com <javascript:>> wrote:
>> > Hi Bas,
>> >
>> > Perhaps, to have this in context, we could add it to e.g. the HoTT web
>> page
>> > and/or the nlab.
>> >
>> > Do you know precise dates for these manuscripts?
>> >
>> > I am looking forward to seeing you in Bonn.
>> >
>> > Also, it would be nice to have Mike Shulman's questions answered or at
>> least
>> > addressed.
>> >
>> > Martin
>> >
>> > On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote:
>> >>
>> >> Hi Martin,
>> >>
>> >> These were discussed publically at some point. I've got them at around
>> >>
>> >> 2000.
>> >> We never put them on the web, because Bishop had decided not to publish
>> >>
>> >> them.
>> >> Since you are doing this now, it might be good to at least add a note
>> >> to that respect, so that people can put them in context.
>> >>
>> >> See you in Bonn!
>> >>
>> >> Bas
>> >>
>> >> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó
>> >> <escar...@gmail.com> wrote:
>> >> > This week I learned two interesting things that seem to be kept as a
>> >> >
>> >> > guarded
>> >> > secret:
>> >> >
>> >> > (1) Errett Bishop reinvented type theory.
>> >> > (2) He also explained how to compile it to Algol.
>> >> >
>> >> > I am adding a link to these two manuscripts. A nice quote from the
>> >> > second
>> >> > paper (Algol.pdf) is this, in my opinion, because it foresees things
>> >> >
>> >> > such as
>> >> > Agda, Coq, NuPrl, ...
>> >> >
>> >> > "The possibility of such a compilation demonstrates the existence of
>> >> >
>> a
>> >> > new
>> >> > type of programming language, one that contains theorems, proofs,
>> >> > quantifications, and implications, in addition to the more
>> conventional
>> >> > facilities for specifying algorithms"
>> >> >
>> >> > This was in the late 1960's (or correct me). Here is a link to both
>> >> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>> >> >
>> >> > Greetings from Bonn.
>> >> > Martin
>> >>
>> > --
>> > You received this message because you are subscribed to the Google
>> Groups
>> > "Homotopy Type Theory" group.
>> > To unsubscribe from this group and stop receiving emails from it, send
>> an
>> > email to HomotopyTypeThe...@googlegroups.com <javascript:>.
>> >
>> > For more options, visit https://groups.google.com/d/optout.
>>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.
>

Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

I have now had a chance to read over the first manuscript more
carefully.  It is quite fascinating!  I think that in modern language,
his system would be called a higher-order logic over a dependent type
theory.  There are some warts from a modern perspective, but I think
it's quite astonishing how close Bishop's system is to modern
dependent type theories and higher-order logics, if in fact there was
historically no communication.

What nowadays we call "types", Bishop calls "classes"; and what we
call "functions" between types he calls "operations".  He has
"power-classes" and "subclasses" which behave roughly like power-types
and sub-types in higher-order logic, along with a separate logic of
formulas that depend on classes.  In particular, propositions are, as
far as I can tell, proof-irrelevant, and *not* identified with types!
He uses the Leibniz equality of HOL (two terms are equal if they
satisfy the same predicates) to formulate the beta and eta rules for
his Pi, Sigma, etc. classes, and includes (p26) the function
extensionality and propositional extensionality axioms again using
this Leibniz equality.

Some other interesting notes about Bishop's system:

1. He has a class of all classes.  I think this means his system is
vulnerable to Girard's paradox and hence inconsistent.  This is
amusing given his remark (p15) that "A contradiction would be just an
indication that we were indulging in meaningless formalism," although
to be fair he also says later (p26) that "If aspects of the
formalization are meaningless, experience will sooner or later let us
know."  Of course, this should be fixable as usual by introducing a
hierarchy of universes.

2. His "sets" (p16) are classes (types) equipped with an equivalence
relation valued in *propositions* (more precisely, equipped with a
subclass of A x A satisfying reflexivity, symmetry, and transitivity).
So they are like setoids defined in Coq with Prop-valued equality
(where Prop satisfies propositional extensionality), not setoids
defined in MLTT with Type-valued equality.

3. He includes the axiom of choice (p12) formulated in terms of his
(proof-irrelevant) propositions, as well as what seems to be a Hilbert
choice operator (though it's not clear to me whether this applies in
open contexts or not).  Since he has powerclasses with propositional
extensionality, I think this means that Diaconescu's argument proves
LEM, which he obviously wouldn't want.  It's harder for me to guess
how this should be fixed, since without some kind of AC, setoids don't
satisfy the principle of unique choice.

4. He makes the class of all sets into a set (p19) with equality
meaning the mere existence of an isomorphism.  But later (p21) he
refers to this set more properly as the set of "cardinal numbers".

5. He also defines a category (p19) to have a class of objects (no
equality relation imposed) and dependent *sets* (classes with equality
relation) of morphisms between any two objects.

6. As we did informally in the HoTT Book, he first introduces
non-dependent function types and then formulates dependent ones (which
he calls "guarded") in terms of a type family expressed as a
non-dependent function into the universe (rather than as a type
expression containing a variable).

It's quite possible, though, that I am misinterpreting some or all of
this; his notation is so different that it's easy to get confused.  If
so, I hope someone will set me straight.



On 5/4/18, Michael Shulman <shu...@sandiego.edu> wrote:
> Right, the question more precisely is whether, when transported along
> whatever isomorphism there is between Bishop's "general language" and
> MLTT (I have not read the manuscript yet to understand this), the
> "sets" defined by Bishop on p16 coincide with Hofmann's setoids.  If
> so, then it would be some substantial additional evidence for the
> claim that setoids are "what Bishop really meant".
>
> On 5/4/18, Martín Hötzel Escardó <escardo...@gmail.com> wrote:
>> (I know that, and probably Mike knows that too. Martin)
>>
>> On Saturday, 5 May 2018 00:12:51 UTC+2, Bas Spitters wrote:
>>>
>>> Setoids were introduced by Martin Hofmann is his PhD-thesis. They were
>>> "inspired" by Bishop; see p8:
>>> www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/ECS-LFCS-95-327.ps
>>>
>>> On Sat, May 5, 2018 at 12:04 AM, Martín Hötzel Escardó
>>> <escar...@gmail.com <javascript:>> wrote:
>>> > Hi Bas,
>>> >
>>> > Perhaps, to have this in context, we could add it to e.g. the HoTT web
>>> page
>>> > and/or the nlab.
>>> >
>>> > Do you know precise dates for these manuscripts?
>>> >
>>> > I am looking forward to seeing you in Bonn.
>>> >
>>> > Also, it would be nice to have Mike Shulman's questions answered or at
>>> least
>>> > addressed.
>>> >
>>> > Martin
>>> >
>>> > On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote:
>>> >>
>>> >> Hi Martin,
>>> >>
>>> >> These were discussed publically at some point. I've got them at
>>> >> around
>>> >>
>>> >> 2000.
>>> >> We never put them on the web, because Bishop had decided not to
>>> >> publish
>>> >>
>>> >> them.
>>> >> Since you are doing this now, it might be good to at least add a note
>>> >> to that respect, so that people can put them in context.
>>> >>
>>> >> See you in Bonn!
>>> >>
>>> >> Bas
>>> >>
>>> >> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó
>>> >> <escar...@gmail.com> wrote:
>>> >> > This week I learned two interesting things that seem to be kept as
>>> >> > a
>>> >> >
>>> >> > guarded
>>> >> > secret:
>>> >> >
>>> >> > (1) Errett Bishop reinvented type theory.
>>> >> > (2) He also explained how to compile it to Algol.
>>> >> >
>>> >> > I am adding a link to these two manuscripts. A nice quote from the
>>> >> > second
>>> >> > paper (Algol.pdf) is this, in my opinion, because it foresees
>>> >> > things
>>> >> >
>>> >> > such as
>>> >> > Agda, Coq, NuPrl, ...
>>> >> >
>>> >> > "The possibility of such a compilation demonstrates the existence
>>> >> > of
>>> >> >
>>> a
>>> >> > new
>>> >> > type of programming language, one that contains theorems, proofs,
>>> >> > quantifications, and implications, in addition to the more
>>> conventional
>>> >> > facilities for specifying algorithms"
>>> >> >
>>> >> > This was in the late 1960's (or correct me). Here is a link to both
>>> >> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>>> >> >
>>> >> > Greetings from Bonn.
>>> >> > Martin
>>> >>
>>> > --
>>> > You received this message because you are subscribed to the Google
>>> Groups
>>> > "Homotopy Type Theory" group.
>>> > To unsubscribe from this group and stop receiving emails from it, send
>>> an
>>> > email to HomotopyTypeThe...@googlegroups.com
>>> > <javascript:>.
>>> >
>>> > For more options, visit https://groups.google.com/d/optout.
>>>
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to HomotopyTypeThe...@googlegroups.com.
>> For more options, visit https://groups.google.com/d/optout.
>>
>

Re: [HoTT] Bishop's work on type theory

[Thorsten Altenkirch]

On 05/05/2018, 05:27, "homotopyt...@googlegroups.com on behalf of
Michael Shulman" <homotopyt...@googlegroups.com on behalf of
shu...@sandiego.edu> wrote:

>3. He includes the axiom of choice (p12) formulated in terms of his
>(proof-irrelevant) propositions, as well as what seems to be a Hilbert
>choice operator (though it's not clear to me whether this applies in
>open contexts or not).  Since he has powerclasses with propositional
>extensionality, I think this means that Diaconescu's argument proves
>LEM, which he obviously wouldn't want.  It's harder for me to guess
>how this should be fixed, since without some kind of AC, setoids don't
>satisfy the principle of unique choice.

Why not? If we identify propositions with setoids that are internally
propositions (all elements are equal) and identify propositions upto
logical equality we get unique choice.

What do I miss here?
Thorsten


>




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Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

I think the problem is that it's not consistent about what a
"proposition" is.  If a "proposition" is a setoid in which all
elements are equal, then to be consistent, the equality relations of
other setoids should also be valued in "propositions" of *this* sort,
not the original collection of "propositions" you started with.
Otherwise, I think you won't necessarily be able to take the quotient
of a setoid by a "proposition"-valued equivalence relation, which is
the whole point of introducing setoids in the first place.  But down
this route lies infinity.

I only know of three ways to get a well-behaved category of setoids:

1. Use propositions as types, as in MLTT Type-valued setoids and the
ex/lex completion.

2. Define a morphism of setoids to be not an operation respecting
equality but a total functional relation, as in the tripos-to-topos
construction and the ex/reg completion.  I personally believe this is
the correct solution in the most generality, but Bishop-style
constructivists don't seem to like it.

3. Assume the axiom of choice, which causes options (1) and (2) to coincide.


On 5/5/18, Thorsten Altenkirch <Thorsten....@nottingham.ac.uk> wrote:
>
>
> On 05/05/2018, 05:27, "homotopyt...@googlegroups.com on behalf of
> Michael Shulman" <homotopyt...@googlegroups.com on behalf of
> shu...@sandiego.edu> wrote:
>
>>3. He includes the axiom of choice (p12) formulated in terms of his
>>(proof-irrelevant) propositions, as well as what seems to be a Hilbert
>>choice operator (though it's not clear to me whether this applies in
>>open contexts or not).  Since he has powerclasses with propositional
>>extensionality, I think this means that Diaconescu's argument proves
>>LEM, which he obviously wouldn't want.  It's harder for me to guess
>>how this should be fixed, since without some kind of AC, setoids don't
>>satisfy the principle of unique choice.
>
> Why not? If we identify propositions with setoids that are internally
> propositions (all elements are equal) and identify propositions upto
> logical equality we get unique choice.
>
> What do I miss here?
> Thorsten
>
>
>>
>
>
>
>
> This message and any attachment are intended solely for the addressee
> and may contain confidential information. If you have received this
> message in error, please contact the sender and delete the email and
> attachment.
>
> Any views or opinions expressed by the author of this email do not
> necessarily reflect the views of the University of Nottingham. Email
> communications with the University of Nottingham may be monitored
> where permitted by law.
>
>
>
>
>

Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

Of course, if you have higher-order logic with propositional
extensionality, you don't need to use setoids at all, but can instead
define quotients as sets of equivalence classes like in ZF.  But I
suspect Bishop wouldn't have liked that either.

On 5/5/18, Michael Shulman <shu...@sandiego.edu> wrote:
> I think the problem is that it's not consistent about what a
> "proposition" is.  If a "proposition" is a setoid in which all
> elements are equal, then to be consistent, the equality relations of
> other setoids should also be valued in "propositions" of *this* sort,
> not the original collection of "propositions" you started with.
> Otherwise, I think you won't necessarily be able to take the quotient
> of a setoid by a "proposition"-valued equivalence relation, which is
> the whole point of introducing setoids in the first place.  But down
> this route lies infinity.
>
> I only know of three ways to get a well-behaved category of setoids:
>
> 1. Use propositions as types, as in MLTT Type-valued setoids and the
> ex/lex completion.
>
> 2. Define a morphism of setoids to be not an operation respecting
> equality but a total functional relation, as in the tripos-to-topos
> construction and the ex/reg completion.  I personally believe this is
> the correct solution in the most generality, but Bishop-style
> constructivists don't seem to like it.
>
> 3. Assume the axiom of choice, which causes options (1) and (2) to
> coincide.
>
>
> On 5/5/18, Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
> wrote:
>>
>>
>> On 05/05/2018, 05:27, "homotopyt...@googlegroups.com on behalf of
>> Michael Shulman" <homotopyt...@googlegroups.com on behalf of
>> shu...@sandiego.edu> wrote:
>>
>>>3. He includes the axiom of choice (p12) formulated in terms of his
>>>(proof-irrelevant) propositions, as well as what seems to be a Hilbert
>>>choice operator (though it's not clear to me whether this applies in
>>>open contexts or not).  Since he has powerclasses with propositional
>>>extensionality, I think this means that Diaconescu's argument proves
>>>LEM, which he obviously wouldn't want.  It's harder for me to guess
>>>how this should be fixed, since without some kind of AC, setoids don't
>>>satisfy the principle of unique choice.
>>
>> Why not? If we identify propositions with setoids that are internally
>> propositions (all elements are equal) and identify propositions upto
>> logical equality we get unique choice.
>>
>> What do I miss here?
>> Thorsten
>>
>>
>>>
>>
>>
>>
>>
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>>
>>
>>
>>
>>
>

Re: [HoTT] Bishop's work on type theory

[Michael Shulman]

On 5/5/18, Michael Shulman <shu...@sandiego.edu> wrote:
> I think the problem is that it's not consistent about what a
> "proposition" is.  If a "proposition" is a setoid in which all
> elements are equal, then to be consistent, the equality relations of
> other setoids should also be valued in "propositions" of *this* sort,
> not the original collection of "propositions" you started with.
> Otherwise, I think you won't necessarily be able to take the quotient
> of a setoid by a "proposition"-valued equivalence relation, which is
> the whole point of introducing setoids in the first place.  But down
> this route lies infinity.

I take that back: this doesn't lead to an infinite sequence, instead
it stops after one iteration, at a result that coincides with (1):

> 1. Use propositions as types, as in MLTT Type-valued setoids and the
> ex/lex completion.

After all, "a proposition is a setoid in which any two elements are
equal" is essentially the same as "a proposition is a type".  This is
of course a perfectly good way to get a category of setoids (although
from a semantic point of view it is too limiting, restricting you to
categories that arise as ex/lex completions), and I believe it's how a
lot of people have interpreted Bishop's mathematics inside MLTT (for
instance).  But unless I'm confused, it's not what Bishop is trying to
do in this manuscript; he definitely seems to want the equality
relations of his setoids to be proof-irrelevant predicates.

Re: [HoTT] Bishop's work on type theory

[Martín Hötzel Escardó]

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Thanks, Mike, for reporting your analysis of the paper. I didn't reply 
earlier because we had a very intensive week with the summer school here at 
the Hausdorff institute. It seems that this is a sort of two-level type 
theory, with a logic on top of a formalism for types.

(One thing that one should always have in mind is that these two papers 
were not published. I have lots of private files with tentative ideas, 
which I wish that if they are ever seen after I die then they will be taken 
as such - tentative ideas.)

I like seeing Bishop offering ideas on what formalism would reflect his 
thinking. But most of all I like his conviction that 

 "The possibility of such a compilation demonstrates the
  existence of a new type of programming language, one that
  contains theorems, proofs, quantifications, and implications,
  in addition to the more conventional facilities for specifying
  algorithms"

as I said before.

I am not sure one can use these two papers as a definitive source to try to 
understand his original, informal "Foundations of constructive analysis". I 
would guess *not*.

Martin


On Saturday, 5 May 2018 06:27:29 UTC+2, Michael Shulman wrote:
>
> I have now had a chance to read over the first manuscript more 
> carefully.  It is quite fascinating!  I think that in modern language, 
> his system would be called a higher-order logic over a dependent type 
> theory.  There are some warts from a modern perspective, but I think 
> it's quite astonishing how close Bishop's system is to modern 
> dependent type theories and higher-order logics, if in fact there was 
> historically no communication. 
>
> What nowadays we call "types", Bishop calls "classes"; and what we 
> call "functions" between types he calls "operations".  He has 
> "power-classes" and "subclasses" which behave roughly like power-types 
> and sub-types in higher-order logic, along with a separate logic of 
> formulas that depend on classes.  In particular, propositions are, as 
> far as I can tell, proof-irrelevant, and *not* identified with types! 
> He uses the Leibniz equality of HOL (two terms are equal if they 
> satisfy the same predicates) to formulate the beta and eta rules for 
> his Pi, Sigma, etc. classes, and includes (p26) the function 
> extensionality and propositional extensionality axioms again using 
> this Leibniz equality. 
>
> Some other interesting notes about Bishop's system: 
>
> 1. He has a class of all classes.  I think this means his system is 
> vulnerable to Girard's paradox and hence inconsistent.  This is 
> amusing given his remark (p15) that "A contradiction would be just an 
> indication that we were indulging in meaningless formalism," although 
> to be fair he also says later (p26) that "If aspects of the 
> formalization are meaningless, experience will sooner or later let us 
> know."  Of course, this should be fixable as usual by introducing a 
> hierarchy of universes. 
>
> 2. His "sets" (p16) are classes (types) equipped with an equivalence 
> relation valued in *propositions* (more precisely, equipped with a 
> subclass of A x A satisfying reflexivity, symmetry, and transitivity). 
> So they are like setoids defined in Coq with Prop-valued equality 
> (where Prop satisfies propositional extensionality), not setoids 
> defined in MLTT with Type-valued equality. 
>
> 3. He includes the axiom of choice (p12) formulated in terms of his 
> (proof-irrelevant) propositions, as well as what seems to be a Hilbert 
> choice operator (though it's not clear to me whether this applies in 
> open contexts or not).  Since he has powerclasses with propositional 
> extensionality, I think this means that Diaconescu's argument proves 
> LEM, which he obviously wouldn't want.  It's harder for me to guess 
> how this should be fixed, since without some kind of AC, setoids don't 
> satisfy the principle of unique choice. 
>
> 4. He makes the class of all sets into a set (p19) with equality 
> meaning the mere existence of an isomorphism.  But later (p21) he 
> refers to this set more properly as the set of "cardinal numbers". 
>
> 5. He also defines a category (p19) to have a class of objects (no 
> equality relation imposed) and dependent *sets* (classes with equality 
> relation) of morphisms between any two objects. 
>
> 6. As we did informally in the HoTT Book, he first introduces 
> non-dependent function types and then formulates dependent ones (which 
> he calls "guarded") in terms of a type family expressed as a 
> non-dependent function into the universe (rather than as a type 
> expression containing a variable). 
>
> It's quite possible, though, that I am misinterpreting some or all of 
> this; his notation is so different that it's easy to get confused.  If 
> so, I hope someone will set me straight. 
>
>
>
> On 5/4/18, Michael Shulman <shu...@sandiego.edu <javascript:>> wrote: 
> > Right, the question more precisely is whether, when transported along 
> > whatever isomorphism there is between Bishop's "general language" and 
> > MLTT (I have not read the manuscript yet to understand this), the 
> > "sets" defined by Bishop on p16 coincide with Hofmann's setoids.  If 
> > so, then it would be some substantial additional evidence for the 
> > claim that setoids are "what Bishop really meant". 
> > 
> > On 5/4/18, Martín Hötzel Escardó <escar...@gmail.com <javascript:>> 
> wrote: 
> >> (I know that, and probably Mike knows that too. Martin) 
> >> 
> >> On Saturday, 5 May 2018 00:12:51 UTC+2, Bas Spitters wrote: 
> >>> 
> >>> Setoids were introduced by Martin Hofmann is his PhD-thesis. They were 
> >>> "inspired" by Bishop; see p8: 
> >>> www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/ECS-LFCS-95-327.ps 
> >>> 
> >>> On Sat, May 5, 2018 at 12:04 AM, Martín Hötzel Escardó 
> >>> <escar...@gmail.com <javascript:>> wrote: 
> >>> > Hi Bas, 
> >>> > 
> >>> > Perhaps, to have this in context, we could add it to e.g. the HoTT 
> web 
> >>> page 
> >>> > and/or the nlab. 
> >>> > 
> >>> > Do you know precise dates for these manuscripts? 
> >>> > 
> >>> > I am looking forward to seeing you in Bonn. 
> >>> > 
> >>> > Also, it would be nice to have Mike Shulman's questions answered or 
> at 
> >>> least 
> >>> > addressed. 
> >>> > 
> >>> > Martin 
> >>> > 
> >>> > On Friday, 4 May 2018 23:57:09 UTC+2, Bas Spitters wrote: 
> >>> >> 
> >>> >> Hi Martin, 
> >>> >> 
> >>> >> These were discussed publically at some point. I've got them at 
> >>> >> around 
> >>> >> 
> >>> >> 2000. 
> >>> >> We never put them on the web, because Bishop had decided not to 
> >>> >> publish 
> >>> >> 
> >>> >> them. 
> >>> >> Since you are doing this now, it might be good to at least add a 
> note 
> >>> >> to that respect, so that people can put them in context. 
> >>> >> 
> >>> >> See you in Bonn! 
> >>> >> 
> >>> >> Bas 
> >>> >> 
> >>> >> On Fri, May 4, 2018 at 11:01 PM, Martín Hötzel Escardó 
> >>> >> <escar...@gmail.com> wrote: 
> >>> >> > This week I learned two interesting things that seem to be kept 
> as 
> >>> >> > a 
> >>> >> > 
> >>> >> > guarded 
> >>> >> > secret: 
> >>> >> > 
> >>> >> > (1) Errett Bishop reinvented type theory. 
> >>> >> > (2) He also explained how to compile it to Algol. 
> >>> >> > 
> >>> >> > I am adding a link to these two manuscripts. A nice quote from 
> the 
> >>> >> > second 
> >>> >> > paper (Algol.pdf) is this, in my opinion, because it foresees 
> >>> >> > things 
> >>> >> > 
> >>> >> > such as 
> >>> >> > Agda, Coq, NuPrl, ... 
> >>> >> > 
> >>> >> > "The possibility of such a compilation demonstrates the existence 
> >>> >> > of 
> >>> >> > 
> >>> a 
> >>> >> > new 
> >>> >> > type of programming language, one that contains theorems, proofs, 
> >>> >> > quantifications, and implications, in addition to the more 
> >>> conventional 
> >>> >> > facilities for specifying algorithms" 
> >>> >> > 
> >>> >> > This was in the late 1960's (or correct me). Here is a link to 
> both 
> >>> >> > manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/ 
> >>> >> > 
> >>> >> > Greetings from Bonn. 
> >>> >> > Martin 
> >>> >> 
> >>> > -- 
> >>> > You received this message because you are subscribed to the Google 
> >>> Groups 
> >>> > "Homotopy Type Theory" group. 
> >>> > To unsubscribe from this group and stop receiving emails from it, 
> send 
> >>> an 
> >>> > email to HomotopyTypeThe...@googlegroups.com 
> <javascript:> 
> >>> > <javascript:>. 
> >>> > 
> >>> > For more options, visit https://groups.google.com/d/optout. 
> >>> 
> >> 
> >> -- 
> >> You received this message because you are subscribed to the Google 
> Groups 
> >> "Homotopy Type Theory" group. 
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>
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> >> 
> > 
>

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Re: [HoTT] Bishop's work on type theory

[Andrej Bauer]

Hi,

I have had these documents in my office for a while, but forgot about
them and never found the time to look at them. Such an opportunity
missed.

In any case, it looks like I have an earlier vesion, because some of
the manuscripts are written in hand in they do look like an incomplete
version of the ones that Martin published. If anyone wants to do
archeological mathematics, I could scan them in, althogh I hesitate to
make them publically available.

With kind regards,

Andrej

Re: Bishop's work on type theory

[Matt Oliveri]

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There doesn't seem to be anything like a conversion rule. I suspect that a 
lot of the math examples developed in the system don't actually type check. 
If they do, it would seem to be luck. Or maybe not; does anyone know some 
key intuition behind this system that I'm missing?

On Friday, May 4, 2018 at 5:01:53 PM UTC-4, Martín Hötzel Escardó wrote:
>
> This week I learned two interesting things that seem to be kept as a 
> guarded secret:
>
> (1) Errett Bishop reinvented type theory.
> (2) He also explained how to compile it to Algol.
>
> I am adding a link to these two manuscripts. A nice quote from the second 
> paper (Algol.pdf) is this, in my opinion, because it foresees things such 
> as Agda, Coq, NuPrl, ...
>
> "The possibility of such a compilation demonstrates the existence of a new 
> type of programming language, one that contains theorems, proofs, 
> quantifications, and implications, in addition to the more conventional 
> facilities for specifying algorithms"
>
> This was in the late 1960's (or correct me). Here is a link to both 
> manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>
> Greetings from Bonn.
> Martin
>

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Re: [HoTT] Re: Bishop's work on type theory

[Michael Shulman]

You mean something like "if t:A and A==B then t:B"?  You're right; I'm
not even sure how to phrase such a rule with Bishop's setup where
"the" type of a term is an *operation* T(t) == A, whereas things like
beta-reduction are expressed only propositionally in terms of Leibniz
equality.

On 5/9/18, Matt Oliveri <atm...@gmail.com> wrote:
> There doesn't seem to be anything like a conversion rule. I suspect that a
> lot of the math examples developed in the system don't actually type check.
>
> If they do, it would seem to be luck. Or maybe not; does anyone know some
> key intuition behind this system that I'm missing?
>
> On Friday, May 4, 2018 at 5:01:53 PM UTC-4, Martín Hötzel Escardó wrote:
>>
>> This week I learned two interesting things that seem to be kept as a
>> guarded secret:
>>
>> (1) Errett Bishop reinvented type theory.
>> (2) He also explained how to compile it to Algol.
>>
>> I am adding a link to these two manuscripts. A nice quote from the second
>>
>> paper (Algol.pdf) is this, in my opinion, because it foresees things such
>>
>> as Agda, Coq, NuPrl, ...
>>
>> "The possibility of such a compilation demonstrates the existence of a new
>>
>> type of programming language, one that contains theorems, proofs,
>> quantifications, and implications, in addition to the more conventional
>> facilities for specifying algorithms"
>>
>> This was in the late 1960's (or correct me). Here is a link to both
>> manuscripts: http://www.cs.bham.ac.uk/~mhe/Bishop/
>>
>> Greetings from Bonn.
>> Martin
>>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
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