Bishop's work on type theory
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Martín Hötzel Escardó
May 4, 2018, 5:01:53 PM5/4/18
to Homotopy Type Theory
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
Michael Shulman
May 4, 2018, 5:19:18 PM5/4/18
to Martín Hötzel Escardó, Homotopy Type Theory
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"?
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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"?
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Bas Spitters
May 4, 2018, 5:57:09 PM5/4/18
to Martín Hötzel Escardó, Homotopy Type Theory
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
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
Martín Hötzel Escardó
May 4, 2018, 6:04:12 PM5/4/18
to Homotopy Type Theory
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
Bas Spitters
May 4, 2018, 6:12:51 PM5/4/18
to Martín Hötzel Escardó, Homotopy Type Theory
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
"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
Martín Hötzel Escardó
May 4, 2018, 6:16:00 PM5/4/18
to Homotopy Type Theory
(I know that, and probably Mike knows that too. Martin)
> email to HomotopyTypeTheory+unsub...@googlegroups.com.
Michael Shulman
May 4, 2018, 6:23:45 PM5/4/18
to Martín Hötzel Escardó, Homotopy Type Theory
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".
>> > email to HomotopyTypeThe...@googlegroups.com <javascript:>.
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".
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Michael Shulman
May 5, 2018, 12:27:29 AM5/5/18
to Martín Hötzel Escardó, Homotopy Type Theory
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.
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.
Thorsten Altenkirch
May 5, 2018, 7:35:55 AM5/5/18
to Michael Shulman, Martín Hötzel Escardó, Homotopy Type Theory
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
>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.
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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Michael Shulman
May 5, 2018, 11:13:54 AM5/5/18
to Thorsten Altenkirch, Martín Hötzel Escardó, Homotopy Type Theory
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.
"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.
Michael Shulman
May 5, 2018, 11:21:17 AM5/5/18
to Thorsten Altenkirch, Martín Hötzel Escardó, Homotopy Type Theory
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.
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.
Michael Shulman
May 5, 2018, 5:27:52 PM5/5/18
to Thorsten Altenkirch, Martín Hötzel Escardó, Homotopy Type Theory
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
> "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.
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.
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.
Matt Oliveri
May 9, 2018, 5:04:01 AM5/9/18
to Homotopy Type Theory
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?
Michael Shulman
May 9, 2018, 12:15:10 PM5/9/18
to Matt Oliveri, Homotopy Type Theory
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.
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.
Martín Hötzel Escardó
May 9, 2018, 6:27:17 PM5/9/18
to Homotopy Type Theory
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
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Andrej Bauer
May 10, 2018, 2:35:37 AM5/10/18
to Homotopy Type Theory
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
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
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