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Rosen, and mapping

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Nick Thompson

Rosen, and mapping

Roseners, and anybody else vaguely interested in category theory.

Rosen seems to be interested in situations in which A maps to B but not all the values in B can be generated by the mapping.

this is a lot like the Intension and the Extension of an utterance. I say with assurance that Mrs. Vanderbilt wished to sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is a function (mathematical sense) that maps from her wants to a subset of the properties of the Titanic. All the properties of the Titanic constitute (in philosophic lingo ) it's extension. The subset, the "image" of Mrs Vanderbilt's wanting , constitutes the intension of her utterance, "I want to sail on the Titanic." Among the titanic's attributes, but outside that image, is the property "hit an iceberg in the North Atlantic and sank."

I guess the question is whether there is a less tortured mathematics than category theory that would allow one to talk about these things.

Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Russell Standish

Re: Rosen, and mapping

The standard language of maps (aka functions) over sets will give you
want you want. Category theory is not needed.

On Sat, Aug 09, 2008 at 08:58:02PM -0600, Nicholas Thompson wrote:

> Roseners, and anybody else vaguely interested in category theory.
>
> Rosen seems to be interested in situations in which A maps to B but not all the values in B can be generated by the mapping.
>
> this is a lot like the Intension and the Extension of an utterance. I say with assurance that Mrs. Vanderbilt wished to sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is a function (mathematical sense) that maps from her wants to a subset of the properties of the Titanic. All the properties of the Titanic constitute (in philosophic lingo ) it's extension. The subset, the "image" of Mrs Vanderbilt's wanting , constitutes the intension of her utterance, "I want to sail on the Titanic." Among the titanic's attributes, but outside that image, is the property "hit an iceberg in the North Atlantic and sank."
>
> I guess the question is whether there is a less tortured mathematics than category theory that would allow one to talk about these things.
>
> N
>
>
>
>
>
> Nicholas S. Thompson
> Emeritus Professor of Psychology and Ethology,
> Clark University ([hidden email])
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

--

----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 [hidden email]
Australia http://www.hpcoders.com.au
----------------------------------------------------------------------------

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Kenneth Lloyd

Re: Rosen, and mapping

In reply to this post by Nick Thompson

Nick,

See: http://math.ucr.edu/home/baez/rosetta.pdf

It relates category theory with mathematical topology, physics, logic and programming.

Ken

From: [hidden email] [mailto:[hidden email]] On Behalf Of Nicholas Thompson
Sent: Saturday, August 09, 2008 8:58 PM
To: [hidden email]
Subject: [FRIAM] Rosen, and mapping

Roseners, and anybody else vaguely interested in category theory.

Rosen seems to be interested in situations in which A maps to B but not all the values in B can be generated by the mapping.

this is a lot like the Intension and the Extension of an utterance. I say with assurance that Mrs. Vanderbilt wished to sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is a function (mathematical sense) that maps from her wants to a subset of the properties of the Titanic. All the properties of the Titanic constitute (in philosophic lingo ) it's extension. The subset, the "image" of Mrs Vanderbilt's wanting , constitutes the intension of her utterance, "I want to sail on the Titanic." Among the titanic's attributes, but outside that image, is the property "hit an iceberg in the North Atlantic and sank."

I guess the question is whether there is a less tortured mathematics than category theory that would allow one to talk about these things.

N

Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

Carl Tollander

Re: Rosen, and mapping

In reply to this post by Russell Standish

Agreed. Nobody convinced me that Rosen was ever really doing category
theory anyhow. If all you need is the category Set, why mobilize
algebraic topology? Leave the hyper-dimensional warp drive in the garage.

Russell Standish wrote:

> The standard language of maps (aka functions) over sets will give you
> want you want. Category theory is not needed.
>
> On Sat, Aug 09, 2008 at 08:58:02PM -0600, Nicholas Thompson wrote:
>
>> Roseners, and anybody else vaguely interested in category theory.
>>
>> Rosen seems to be interested in situations in which A maps to B but not all the values in B can be generated by the mapping.
>>
>> this is a lot like the Intension and the Extension of an utterance. I say with assurance that Mrs. Vanderbilt wished to sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is a function (mathematical sense) that maps from her wants to a subset of the properties of the Titanic. All the properties of the Titanic constitute (in philosophic lingo ) it's extension. The subset, the "image" of Mrs Vanderbilt's wanting , constitutes the intension of her utterance, "I want to sail on the Titanic." Among the titanic's attributes, but outside that image, is the property "hit an iceberg in the North Atlantic and sank."
>>
>> I guess the question is whether there is a less tortured mathematics than category theory that would allow one to talk about these things.
>>
>> N
>>
>>
>>
>>
>>
>> Nicholas S. Thompson
>> Emeritus Professor of Psychology and Ethology,
>> Clark University ([hidden email])
>> ============================================================
>> FRIAM Applied Complexity Group listserv
>> Meets Fridays 9a-11:30 at cafe at St. John's College
>> lectures, archives, unsubscribe, maps at http://www.friam.org
>>
>
>

Phil Henshaw-2

Re: Rosen, and mapping

In reply to this post by Nick Thompson

It seems Rosen would be concerned with incomplete mapping for categories of things in relation to categories of reason. Take the ideal condition: assume that nature is completely consistent with her categories and people are perfectly self-consistent in using theirs, will it then be possible to arrange a correspondence between the two?

From: [hidden email] [mailto:[hidden email]] On Behalf Of Nicholas Thompson
Sent: Saturday, August 09, 2008 10:58 PM
To: [hidden email]
Subject: [FRIAM] Rosen, and mapping

Roseners, and anybody else vaguely interested in category theory.

Rosen seems to be interested in situations in which A maps to B but not all the values in B can be generated by the mapping.

I guess the question is whether there is a less tortured mathematics than category theory that would allow one to talk about these things.

Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

Roger Frye-3

Re: Rosen, and mapping

In reply to this post by Russell Standish

I agree with Russell and Carl, but a couple of mathematical examples
might help.

Consider the mapping (i.e. arrow) from a pair of factors to their
product. There is not a unique reverse mapping from the product to
the factors. Also, if the factors are positive, consider the mapping
from them to their individual logarithms; then a mapping from that
pair to their sum. The logarithm and anti-logarithm provide a two
directional arrow between the sum and product, allowing sums of
logarithms to be used in place of multiplication.

Andrew Wiles summarized the problem of Fermat's Last Theorem as
knowing that there were arrows in one direction between elliptic
curves, modular forms and galois fields, but needing to show that one
of the arrows could be reversed for the particular elliptic curve that
represented a^n+b^n=c^n for n>5.
-Roger

On Aug 9, 2008, at 9:14 PM, Russell Standish wrote:

> The standard language of maps (aka functions) over sets will give you
> want you want. Category theory is not needed.
>
> On Sat, Aug 09, 2008 at 08:58:02PM -0600, Nicholas Thompson wrote:
>> Roseners, and anybody else vaguely interested in category theory.
>>
>> Rosen seems to be interested in situations in which A maps to B but
>> not all the values in B can be generated by the mapping.
>>
>> this is a lot like the Intension and the Extension of an
>> utterance. I say with assurance that Mrs. Vanderbilt wished to
>> sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is a
>> function (mathematical sense) that maps from her wants to a subset
>> of the properties of the Titanic. All the properties of the
>> Titanic constitute (in philosophic lingo ) it's extension. The
>> subset, the "image" of Mrs Vanderbilt's wanting , constitutes the
>> intension of her utterance, "I want to sail on the Titanic." Among
>> the titanic's attributes, but outside that image, is the property
>> "hit an iceberg in the North Atlantic and sank."
>>
>> I guess the question is whether there is a less tortured
>> mathematics than category theory that would allow one to talk about
>> these things.
>>

Kenneth Lloyd

Re: Rosen, and mapping

The physical realization of this mathematics is described in statistical
mechanics and non-equilibrium thermodynamics by Prigogine and the
Brussels-Austin group.

Ken

> -----Original Message-----
> From: [hidden email]
> [mailto:[hidden email]] On Behalf Of Roger Frye
> Sent: Sunday, August 10, 2008 7:24 AM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: Re: [FRIAM] Rosen, and mapping
>
> I agree with Russell and Carl, but a couple of mathematical
> examples might help.
>
> Consider the mapping (i.e. arrow) from a pair of factors to
> their product. There is not a unique reverse mapping from
> the product to the factors. Also, if the factors are
> positive, consider the mapping from them to their individual
> logarithms; then a mapping from that pair to their sum. The
> logarithm and anti-logarithm provide a two directional arrow
> between the sum and product, allowing sums of logarithms to
> be used in place of multiplication.
>
> Andrew Wiles summarized the problem of Fermat's Last Theorem
> as knowing that there were arrows in one direction between
> elliptic curves, modular forms and galois fields, but needing
> to show that one of the arrows could be reversed for the
> particular elliptic curve that represented a^n+b^n=c^n for n>5.
> -Roger
>
> On Aug 9, 2008, at 9:14 PM, Russell Standish wrote:
>
> > The standard language of maps (aka functions) over sets
> will give you
> > want you want. Category theory is not needed.
> >
> > On Sat, Aug 09, 2008 at 08:58:02PM -0600, Nicholas Thompson wrote:
> >> Roseners, and anybody else vaguely interested in category theory.
> >>
> >> Rosen seems to be interested in situations in which A maps
> to B but
> >> not all the values in B can be generated by the mapping.
> >>
> >> this is a lot like the Intension and the Extension of an
> utterance.
> >> I say with assurance that Mrs. Vanderbilt wished to sail on the
> >> Titanic. In this case, Mrs Vanderbilt's "wanting" is a function
> >> (mathematical sense) that maps from her wants to a subset of the
> >> properties of the Titanic. All the properties of the Titanic
> >> constitute (in philosophic lingo ) it's extension. The
> subset, the
> >> "image" of Mrs Vanderbilt's wanting , constitutes the intension of
> >> her utterance, "I want to sail on the Titanic." Among the
> titanic's
> >> attributes, but outside that image, is the property "hit
> an iceberg
> >> in the North Atlantic and sank."
> >>
> >> I guess the question is whether there is a less tortured
> mathematics
> >> than category theory that would allow one to talk about
> these things.
> >>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

Marcus G. Daniels

Re: Rosen, and mapping

In reply to this post by Carl Tollander

One contribution from category theory for dealing with stateful systems
(like organisms) is the Monad.
Monads provide a way to compose together computations into larger ones
such that an order of execution can be enforced *and* such that the
state doesn't need to be passed around from amongst the functions.
Without a design pattern like this, it isn't possible to talk about the
interactions of a complex set of objects and their internal changes
without allowing side-effects (which will mean the functions aren't
really functions in the mathematical sense), or exposing irrelevant and
often complex internal state in the equation parameters.

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Carl Tollander

Re: Rosen, and mapping

In reply to this post by Kenneth Lloyd

In that same vein I also recommend at least the first few pages of this
talk.

http://math.ucr.edu/home/baez/periodic/

Carl

Ken Lloyd wrote:

> Nick,
>
> See: http://math.ucr.edu/home/baez/rosetta.pdf
>
> It relates category theory with mathematical topology, physics, logic
> and programming.
>
> Ken
>
> *From:* [hidden email]
> [mailto:[hidden email]] *On Behalf Of *Nicholas Thompson
> *Sent:* Saturday, August 09, 2008 8:58 PM
> *To:* [hidden email]
> *Subject:* [FRIAM] Rosen, and mapping
>
> Roseners, and anybody else vaguely interested in category theory.
>
> Rosen seems to be interested in situations in which A maps to B
> but not all the values in B can be generated by the mapping.
>
> this is a lot like the Intension and the Extension of an
> utterance. I say with assurance that Mrs. Vanderbilt wished to
> sail on the Titanic. In this case, Mrs Vanderbilt's "wanting" is
> a function (mathematical sense) that maps from her wants to a
> subset of the properties of the Titanic. All the properties of
> the Titanic constitute (in philosophic lingo ) it's extension.
> The subset, the "image" of Mrs Vanderbilt's wanting , constitutes
> the intension of her utterance, "I want to sail on the Titanic."
> Among the titanic's attributes, but outside that image, is the
> property "hit an iceberg in the North Atlantic and sank."
>
> I guess the question is whether there is a less tortured
> mathematics than category theory that would allow one to talk
> about these things.
>
> N
>
>
>
>
> Nicholas S. Thompson
> Emeritus Professor of Psychology and Ethology,
> Clark University ([hidden email] <mailto:[hidden email]>)
>
>
>
>
> ------------------------------------------------------------------------
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

Kenneth Lloyd

Re: Rosen, and mapping

Carl,

Notice that Baez and Stay are (or have been) working with Bob Coecke of
Oxford (New Structures in Physics). See: Bob Coecke's work with Samson
Abramsky on the applicability of this work to dynamic and quantum logic.

Ken

> -----Original Message-----
> From: [hidden email]
> [mailto:[hidden email]] On Behalf Of Carl Tollander
> Sent: Sunday, August 10, 2008 10:14 AM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: Re: [FRIAM] Rosen, and mapping
>
> In that same vein I also recommend at least the first few
> pages of this talk.
>
> http://math.ucr.edu/home/baez/periodic/
>
> Carl
>
> Ken Lloyd wrote:
> > Nick,
> >
> > See: http://math.ucr.edu/home/baez/rosetta.pdf
> >
> > It relates category theory with mathematical topology,
> physics, logic
> > and programming.
> >
> > Ken
> >
> > *From:* [hidden email]
> > [mailto:[hidden email]] *On Behalf Of
> *Nicholas Thompson
> > *Sent:* Saturday, August 09, 2008 8:58 PM
> > *To:* [hidden email]
> > *Subject:* [FRIAM] Rosen, and mapping
> >
> > Roseners, and anybody else vaguely interested in
> category theory.
> >
> > Rosen seems to be interested in situations in which A maps to B
> > but not all the values in B can be generated by the mapping.
> >
> > this is a lot like the Intension and the Extension of an
> > utterance. I say with assurance that Mrs. Vanderbilt wished to
> > sail on the Titanic. In this case, Mrs Vanderbilt's
> "wanting" is
> > a function (mathematical sense) that maps from her wants to a
> > subset of the properties of the Titanic. All the properties of
> > the Titanic constitute (in philosophic lingo ) it's extension.
> > The subset, the "image" of Mrs Vanderbilt's wanting ,
> constitutes
> > the intension of her utterance, "I want to sail on the
> Titanic."
> > Among the titanic's attributes, but outside that image, is the
> > property "hit an iceberg in the North Atlantic and sank."
> >
> > I guess the question is whether there is a less tortured
> > mathematics than category theory that would allow one to talk
> > about these things.
> >
> > N
> >
> >
> >
> >
> > Nicholas S. Thompson
> > Emeritus Professor of Psychology and Ethology,
> > Clark University ([hidden email]
> > <mailto:[hidden email]>)
> >
> >
> >
> >
> >
> ----------------------------------------------------------------------
> > --
> >
> > ============================================================
> > FRIAM Applied Complexity Group listserv Meets Fridays
> 9a-11:30 at cafe
> > at St. John's College lectures, archives, unsubscribe, maps at
> > http://www.friam.org
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

Carl Tollander

Re: ..., and mapping

Ken,

Yah, brief Wedtech conversation on Abramsky March last year.

http://web.comlab.ox.ac.uk/oucl/work/samson.abramsky/bertinoro05.pdf

Actually I think it was in terms of a "world's best abstract"
conversation, but fun nonetheless.

Anyhow, not enough air in the sails at the time to follow through.
Found from Dantas blog post on some quantum concurrency ideas and now I
see Coecke occasionally has some entries over at N-Category Cafe.
Maybe it's time to revisit now that the Sander Bais talk last week
rekindled my interest in QC.

Carl

Ken Lloyd wrote:

> Carl,
>
> Notice that Baez and Stay are (or have been) working with Bob Coecke of
> Oxford (New Structures in Physics). See: Bob Coecke's work with Samson
> Abramsky on the applicability of this work to dynamic and quantum logic.
>
> Ken
>
>
>
>

Phil Henshaw-2

Re: Rosen, and mapping

In reply to this post by Marcus G. Daniels

So far I only hear issues about mapping theoretical things to theoretical
things, math to math, theory to theory. Last I knew the only mapping
between physical and theoretical things had to do with ranges of uncertainty
in measures of the physical things, like weight and height guessing, and
that critical step of abstracting a measure from physical things as the
basis of correspondence, to give math something to correlate, was completely
necessary. Have you guys dispensed with that somehow?

Phil

> -----Original Message-----
> From: [hidden email] [mailto:[hidden email]] On
> Behalf Of Marcus G. Daniels
> Sent: Sunday, August 10, 2008 11:51 AM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: Re: [FRIAM] Rosen, and mapping
>
> One contribution from category theory for dealing with stateful systems
> (like organisms) is the Monad.
> Monads provide a way to compose together computations into larger ones
> such that an order of execution can be enforced *and* such that the
> state doesn't need to be passed around from amongst the functions.
> Without a design pattern like this, it isn't possible to talk about the
> interactions of a complex set of objects and their internal changes
> without allowing side-effects (which will mean the functions aren't
> really functions in the mathematical sense), or exposing irrelevant and
> often complex internal state in the equation parameters.
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org