Friam

Isomorphism between computation and philosophy

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Owen Densmore

Isomorphism between computation and philosophy

Administrator

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:

Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

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Meets Fridays 9a-11:30 at cafe at St. John's College
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doug carmichael

Re: Isomorphism between computation and philosophy

Philosophy is very broad and includes many things like ethics and anesthetics. A good test case would be not logic, but poetry.

Blessings,

Doug

http://dougcarmichael.com

http://gardenworldpolitics.com

On Apr 16, 2013, at 9:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Nick Thompson

Re: Isomorphism between computation and philosophy

In reply to this post by Owen Densmore

Well, first, Owen, you should know that I am not a philosopher, but more a philosopher groupie, drawn to philosophers by their ability to help me be coherent … that is, to help me make the thoughts in one part of my thinking consistent with my thoughts in others.

In my teaching career, I have always proceeded on the (Pragmatist) assumption that if people talk long enough they will come to an understanding of one another. But … I shudder to admit it … I am beginning to think that this is a pipe dream. Actually, I have come a long way to your view about philosophy … that philosophers aren’t really trying to come to terms and that – worse – it is perhaps impossible for them to do so. That is, while philosophy might be helpful in pointing out incoherences in my thought, philosophers are not dedicated to becoming coherent amongst themselves. I am not at all sure how I could go back to teaching having come to these conclusions. It’s probably a good thing that I retired before I got wise.

Translatability has been a crucial issue in modern analytical philosophy. Translation implies that you and I have the same piano and that, while we may call the keys by different names, there is a key on your piano that corresponds to every key on mine. But philosophers have more or less given up on translateablity, I think.

Still, I am tempted to start with the assumption that there is a word, or small group of words, in my vocabulary that corresponds to your word, undecideable. Can you guess at what those words might be?

Nick

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 10:26 AM
To: Complexity Coffee Group
Subject: [FRIAM] Isomorphism between computation and philosophy

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:

Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

lrudolph

Re: Isomorphism between computation and philosophy

Nick:

> It's probably a good thing that I retired before I got wise.

I think I hear the sound of the Arrow of Causality twanging in the bullseye.

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Meets Fridays 9a-11:30 at cafe at St. John's College
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Barry MacKichan

Re: Isomorphism between computation and philosophy

In reply to this post by Owen Densmore

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable.

The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.)

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Owen Densmore

Re: Isomorphism between computation and philosophy

Administrator

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

To tell the truth, Godel .. it seems to me .. says to the practicing mathematician that the axioms have to be very carefully chosen. Its sorta like linear algebra: a system can be over constrained .. thus contain impossibilities, or under constrained thus have multiple solutions.

But all I'm hoping for is any attempt to make the words Nick and others have be as precise as a computer language. If this is the case, then we can use the lovely computation hierarchy from FSA, to CFL to Turing/Church. But then, most mathematicians know none of this structure either. Sigh.

I wish philosophy had the same constraints where bugs could be found. On the other hand, ambiguity can be a huge plus, as any spoken language shows.

-- Owen

On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan <[hidden email]> wrote:

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable.

The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.)

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:

Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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Nick Thompson

Re: Isomorphism between computation and philosophy

Owen,

One of the reasons that mathematical language can be so precise is that it isn’t ABOUT anything, right? The minute one adds semantics …. the minute one applies mathematics to anything … all the problems of ordinary language begin to manifest themselves, don’t they?

Nick

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 3:50 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

I wish philosophy had the same constraints where bugs could be found. On the other hand, ambiguity can be a huge plus, as any spoken language shows.

-- Owen

On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan <[hidden email]> wrote:

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Barry MacKichan

Re: Isomorphism between computation and philosophy

In reply to this post by Barry MacKichan

I should correct myself. The mapping is not necessarily an isomorphism.

--Barry

On Apr 16, 2013, at 3:39 PM, Barry MacKichan <[hidden email]> wrote:

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable.

The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.)

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Barry MacKichan

Re: Isomorphism between computation and philosophy

In reply to this post by Nick Thompson

They say all mathematicians are Platonists. The interesting thing is that this thing that is not about anything can be so surprising.

--Barry

On Apr 16, 2013, at 3:53 PM, "Nicholas Thompson" <[hidden email]> wrote:

Owen,

One of the reasons that mathematical language can be so precise is that it isn’t ABOUT anything, right? The minute one adds semantics …. the minute one applies mathematics to anything … all the problems of ordinary language begin to manifest themselves, don’t they?

Nick

From: Friam [mailto:friam-[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 3:50 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

To tell the truth, Godel .. it seems to me .. says to the practicing mathematician that the axioms have to be very carefully chosen. Its sorta like linear algebra: a system can be over constrained .. thus contain impossibilities, or under constrained thus have multiple solutions.

But all I'm hoping for is any attempt to make the words Nick and others have be as precise as a computer language. If this is the case, then we can use the lovely computation hierarchy from FSA, to CFL to Turing/Church. But then, most mathematicians know none of this structure either. Sigh.

I wish philosophy had the same constraints where bugs could be found. On the other hand, ambiguity can be a huge plus, as any spoken language shows.

-- Owen

On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan <[hidden email]> wrote:
Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable.

The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.)

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Barry MacKichan

Re: Isomorphism between computation and philosophy

In reply to this post by Owen Densmore

Actually, Godel said "that the axioms [have to]->[can't] be very carefully chosen." The theorem says that any mathematical system that contains the integers cannot be both complete and self-consistent. It is unique in the list of 'impossibility' theorems in that it has a mathematical proof. The others in your list are all contingent on some form of observation.

It's sort of like saying all sets of equations have to be overdetermined or underdetermined or both. Except its really hits at the roots of the mathematical enterprise. They say its announcement hit Bertrand Russell really hard.

-Barry

On Apr 16, 2013, at 3:49 PM, Owen Densmore <[hidden email]> wrote:

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

To tell the truth, Godel .. it seems to me .. says to the practicing mathematician that the axioms have to be very carefully chosen.

Owen Densmore

Re: Isomorphism between computation and philosophy

Administrator

In reply to this post by Nick Thompson

No, I think we can make a mapping from mathematical concepts to things. Integers, for example, can be made to map onto any discrete semantic concept. At the simplest level, we can nicely define an atom. We can make a countable mapping onto them (note: countable can be finite). There's lots of atoms, but mathematics comfortably manages.

Similarly, computers are concrete things. We have a fine mathematics for computational devices, a hierarchy of devices: Finite State Automata, Context Free Languages, and Turring Machines. They all have equivalent, somewhat more powerful, devices like the Non Deterministic Finite Automata set which can all be reduced to FSAs.

This is pretty concrete: we can with extreme confidence discuss what these machines can do and classify programs that can or cannot be implemented by them.

More properly, we can discuss inputs to devices as "alphabets over symbol sets". We can define the accepting states of the device, thus equivalently the substrings of the alphabets that are accepted by the device. We can also define our devices quite clearly.

For example, the FSA is a 5-tuple (Q, S, d, q0, F) where Q are a finite set of states, S is the finite set of symbols, the alphabet, d is a delta function which given a symbol and a state yields a next state, q0 is the start state, and F is a subset of Q which "accept" the input string. The set of strings that end up at F are called the "language" of the device.

These are both abstract and concrete. But given an alphabet and a FSA 5-tuple, I can prove things about the inputs and outputs. In particular, given an alphabet of {0,1} I can prove that there is no FSA that can accept the language of n-0s followed by exactly n-1's where n can be arbitrary but finite. In other words, I can prove a FSA cannot "count".

Briefly, we can also show that the higher device level, the Turing Machine, has similar limits. The proof is fairly simple, proving that the languages of a TM is the continuum while the number of inputs is countable infinite. Thus there are members of the languages that a TM could accept that are outside of the countable computations of a TM.

So there's stuff we can't compute.

The joy of the symbolic/axiomatic approach is not that it is free of semantics, but that we can devise ways to map math to real things.

I doubt you would say this does not mean anything.

-- Owen

On Tue, Apr 16, 2013 at 3:53 PM, Nicholas Thompson <[hidden email]> wrote:

Owen,

One of the reasons that mathematical language can be so precise is that it isn’t ABOUT anything, right? The minute one adds semantics …. the minute one applies mathematics to anything … all the problems of ordinary language begin to manifest themselves, don’t they?

Nick

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 3:50 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

To tell the truth, Godel .. it seems to me .. says to the practicing mathematician that the axioms have to be very carefully chosen. Its sorta like linear algebra: a system can be over constrained .. thus contain impossibilities, or under constrained thus have multiple solutions.

But all I'm hoping for is any attempt to make the words Nick and others have be as precise as a computer language. If this is the case, then we can use the lovely computation hierarchy from FSA, to CFL to Turing/Church. But then, most mathematicians know none of this structure either. Sigh.

I wish philosophy had the same constraints where bugs could be found. On the other hand, ambiguity can be a huge plus, as any spoken language shows.

-- Owen

On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan <[hidden email]> wrote:

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable.

The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.)

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:

Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Owen Densmore

Re: Isomorphism between computation and philosophy

Administrator

In reply to this post by Barry MacKichan

Arrow's impossibility theorem is provable, basically social choice is impossible given several fairly sound requirements: 3 or more things to choose between and transitivity of choice.

C isn't a proof, agreed. Although its acceptance is well seen by observation. And physics hasn't theorems in the same sense as mathematics. Bad choice on my part.

Heisenberg is directly provable from Schrödinger's equation

Decidability is provable by showing the acceptance set of TMs is countably infinite while the possible languages is continuously infinite (integers vs reals)

NoFreeLunch simply shows that random methods (GAs etc) have inputs that are no better managed than uniformly random guessing. But fortunately, the pessimal inputs are rare and NFL did us the favor of finding where to look for tractable stochastic algorithms. Whew!

On Tue, Apr 16, 2013 at 4:13 PM, Barry MacKichan <[hidden email]> wrote:

Actually, Godel said "that the axioms [have to]->[can't] be very carefully chosen." The theorem says that any mathematical system that contains the integers cannot be both complete and self-consistent. It is unique in the list of 'impossibility' theorems in that it has a mathematical proof. The others in your list are all contingent on some form of observation.

It's sort of like saying all sets of equations have to be overdetermined or underdetermined or both. Except its really hits at the roots of the mathematical enterprise. They say its announcement hit Bertrand Russell really hard.

-Barry

On Apr 16, 2013, at 3:49 PM, Owen Densmore <[hidden email]> wrote:

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

To tell the truth, Godel .. it seems to me .. says to the practicing mathematician that the axioms have to be very carefully chosen.

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Nick Thompson

Re: Isomorphism between computation and philosophy

In reply to this post by Owen Densmore

I don’t think I said that math couldn’t be mapped onto things. I only said that such mappings are not essential to math and, further, that when such mappings occur, the door is opened for confusion that is opened in any semantic relation.

Barry will have to handle the rest of what you said.

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 5:12 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

This is pretty concrete: we can with extreme confidence discuss what these machines can do and classify programs that can or cannot be implemented by them.

So there's stuff we can't compute.

The joy of the symbolic/axiomatic approach is not that it is free of semantics, but that we can devise ways to map math to real things.

I doubt you would say this does not mean anything.

-- Owen

On Tue, Apr 16, 2013 at 3:53 PM, Nicholas Thompson <[hidden email]> wrote:

Owen,

Nick

One has to be careful with nearly all the "impossibility" theorems: Arrow's voting, the speed of light, Godel, Heisenberg, decidability, NoFreeLunch, ... and so on.

I wish philosophy had the same constraints where bugs could be found. On the other hand, ambiguity can be a huge plus, as any spoken language shows.

-- Owen

On Tue, Apr 16, 2013 at 3:39 PM, Barry MacKichan <[hidden email]> wrote:

Curious. Isn't the proof of Godel's theorem a special case of this?

As I understand it, the proof is this:

Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context.

--Barry

On Apr 16, 2013, at 10:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen
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Owen Densmore

Re: Isomorphism between computation and philosophy

Administrator

On Tue, Apr 16, 2013 at 6:10 PM, Nicholas Thompson <[hidden email]> wrote:

I don’t think I said that math couldn’t be mapped onto things. I only said that such mappings are not essential to math and, further, that when such mappings occur, the door is opened for confusion that is opened in any semantic relation.

Could you show me such a thing? I demonstrated that computers for example do not suffer from this confusion. Computing is a branch of mathematics that looked inward and found it could provide real world mappings from 5-tuples defining a computing engine (the FSA) to real computers. Every time you step on the in/out mat for a door at a store, you are implementing a FSA. (Note I bow to your "door" above :)

Call it "Applied Mathematics" if you'd prefer. But it certainly has a very high reality coefficient. There is no ambiguity and there is semantic binding.

(Note: I realize that ABM does deal with this, and we've dealt with it with your MOTH model, but it is not necessarily general.)

Let me simplify. Is there a realm in which philosophy can exhibit a bug? And more specifically by simply "running" the philosophy engine?

I believe this may be possible, but I'm not sure. Maybe we'd have to create a new field. Certainly Turing, Church, von Neumann, Shannon, and many other in the computational world did. They stood on a brink, vital for going forward. Von Neumann had to argue for a computer to be admitted to the Institute for Advanced Study in Princeton .. it was considered just a machine. Church and Turing showed that to be nonsense. Can we do the same for philosophy?

NB: I'm not referring to "computational complexity" in which we deal with the running time issues of an algorithm, but to the semantics of computation itself. We really do have a strong grasp on what computation is and we do not quibble about meaning .. at least without heading immediately to axiomatic solutions.

-- Owen

Nick Thompson

Re: Isomorphism between computation and philosophy

Oh, gosh, owen. I am trying to think of somebody to forward this on to. Dennett would be the obvious guy, but he only rarely answers my mail.

Eric, can you think of somebody in your acquaintance who would be willing to comment on reference always introduces ambiguity, or whether there is an in principle distinction between applied math and philosophical argument.

Nick

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, April 16, 2013 9:12 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

On Tue, Apr 16, 2013 at 6:10 PM, Nicholas Thompson <[hidden email]> wrote:

I don’t think I said that math couldn’t be mapped onto things. I only said that such mappings are not essential to math and, further, that when such mappings occur, the door is opened for confusion that is opened in any semantic relation.

Call it "Applied Mathematics" if you'd prefer. But it certainly has a very high reality coefficient. There is no ambiguity and there is semantic binding.

(Note: I realize that ABM does deal with this, and we've dealt with it with your MOTH model, but it is not necessarily general.)

Let me simplify. Is there a realm in which philosophy can exhibit a bug? And more specifically by simply "running" the philosophy engine?

-- Owen

Steve Smith

Re: Isomorphism between computation and philosophy

In reply to this post by doug carmichael

Doug -

Thanks for weighing in here... as an aside, I skimmed "Garden World" and found it compelling... I hope others here will take the time!

On the thread topic, it would be rather "convenient" in many ways if there were such an isomorphism as Owen seeks (postulates), but I find it to reflect a fundamental misunderstanding of "what is knowledge"?

Other parts of the thread, relating to the question of semantics begins to address this. Intuitively, it is like thinking that one can make visual art without awareness of the negative space and the context it exists in, or of writing poetry (or really anything but the driest of prose as well?) without appreciating that it much of what is being said is "between the lines".

I have a friend who wrote a program to parse and analyze the logic in Aquinas' Summa Theologica and claimed to find numerous (but not outrageous) simple errors in his logic. That isn't in any way close to imagining that one could translate such a text into symbolic logic and determine anything (else) more significant from it than internal consistency and/or consistency with some external axiomatic system.

- Steve

Philosophy is very broad and includes many things like ethics and anesthetics. A good test case would be not logic, but poetry.

Blessings,
Doug

http://dougcarmichael.com

http://gardenworldpolitics.com

On Apr 16, 2013, at 9:25 AM, Owen Densmore <[hidden email]> wrote:

On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:

Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
============================================================
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Meets Fridays 9a-11:30 at cafe at St. John's College
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Nick Thompson

Re: Isomorphism between computation and philosophy

I thought Bertrand Russell WAS the isomorphism between philosophy and computation!

Ach! Aging isn’t for weaklings.

From: Friam [mailto:[hidden email]] On Behalf Of Steve Smith
Sent: Tuesday, April 16, 2013 10:30 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

Philosophy is very broad and includes many things like ethics and anesthetics. A good test case would be not logic, but poetry.

Blessings,
Doug
http://dougcarmichael.com
http://gardenworldpolitics.com

On Apr 16, 2013, at 9:25 AM, Owen Densmore <[hidden email]> wrote:
On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <[hidden email]> wrote:
Can anybody translate this for a non programmer person?

Nick's question brings up a project I'd love to see: an attempt at an isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, onto mapping from one to another, or a bijection.)

For example, in computer science, "decidability" is a very concrete idea. Yet when I hear philosophical terms, and dutifully look them up in the stanford dictionary of philosophy, I find myself suspicious of circularity.

Decidability is interesting because it proves not all computations can successfully expressed as "programs". It does this by using two infinities of different cardinality (countable vs continuum).

Does philosophy deal in constructs that nicely map onto computing, possibly programming languages?

I'm not specifically concerned with decidability, only use that as an example because it shows the struggle in computer science for modeling computation itself, from Finite Automata, Context Free Languages, and to Turing Machines (or equivalently lambda calculus).

I don't dislike philosophy, mainly thanks to conversations with Nick. And I do know that axiomatic approaches to philosophy have been popular.

So is there a possible isomorphism?

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

Owen Densmore

Re: Isomorphism between computation and philosophy

Administrator

In reply to this post by Nick Thompson

On Tue, Apr 16, 2013 at 11:10 AM, Nicholas Thompson <[hidden email]> wrote:

<snip>

Translatability has been a crucial issue in modern analytical philosophy. Translation implies that you and I have the same piano and that, while we may call the keys by different names, there is a key on your piano that corresponds to every key on mine. But philosophers have more or less given up on translateablity, I think.

That seems like a useful concept. Why did they give up on it?

Still, I am tempted to start with the assumption that there is a word, or small group of words, in my vocabulary that corresponds to your word, undecideable. Can you guess at what those words might be?

Interestingly enough, the stanford encyclopedia of philosophy has decidability all over the place, so maybe (un)decidable is a reasonably good philosophical concept already. They use it in basically the same way computing folk do. But then Frank tells me that the philosophy departments are using highly specialized mathematics.

Unfortunately, if an area of philosophy is undecidable, it has a "halting problem" .. i.e. no sense discussing it any further! :)

Nick

Nick Thompson

Re: Isomorphism between computation and philosophy

Owen,

I wish we could drag Frank into this conversation, because he is the only person we know who stands firmly in both worlds.

So, Owen, you meet a beautiful woman at a cocktail party. She seems intelligent, not a person to be fobbed off, but has no experience with either Maths or Computer Science. She looks deep into your eyes, and asks “And what, Mr. Densmore, is the halting problem?” You find yourself torn between two impulses. One is to use the language that would give you credibility in the world of your mentors and colleagues. But you realize that that language is going to be of absolutely no use to her, however ever much it might make you feel authoritative to use it. She expects an answer. Yet you hesitate. What language do you use?

You would start, would you not, with the idea of a “problem.” A problem is some sort of difficulty that needs to be surmounted. There is a goal and something that thwarts that goal. What are these elements in the halting PROBLEM? And why is HALTING a problem?

Nick

From: Friam [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Wednesday, April 17, 2013 9:09 AM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Isomorphism between computation and philosophy

On Tue, Apr 16, 2013 at 11:10 AM, Nicholas Thompson <[hidden email]> wrote:

<snip>

Translatability has been a crucial issue in modern analytical philosophy. Translation implies that you and I have the same piano and that, while we may call the keys by different names, there is a key on your piano that corresponds to every key on mine. But philosophers have more or less given up on translateablity, I think.

That seems like a useful concept. Why did they give up on it?

Still, I am tempted to start with the assumption that there is a word, or small group of words, in my vocabulary that corresponds to your word, undecideable. Can you guess at what those words might be?

Unfortunately, if an area of philosophy is undecidable, it has a "halting problem" .. i.e. no sense discussing it any further! :)

Nick

Barry MacKichan

Re: Isomorphism between computation and philosophy

In reply to this post by Owen Densmore

On a tangential note, I was told in 1961 of a project to prove (on a computer) the theorems in Principia Mathematica. It went well through the first section, and then they hit the brick wall when they encountered statements like "there exists" and "for every". When dealing with infinite sets, these can be hard.

On Apr 16, 2013, at 9:12 PM, Owen Densmore <[hidden email]> wrote:

On Tue, Apr 16, 2013 at 6:10 PM, Nicholas Thompson <[hidden email]> wrote:

I don’t think I said that math couldn’t be mapped onto things. I only said that such mappings are not essential to math and, further, that when such mappings occur, the door is opened for confusion that is opened in any semantic relation.

Could you show me such a thing? I demonstrated that computers for example do not suffer from this confusion. Computing is a branch of mathematics that looked inward and found it could provide real world mappings from 5-tuples defining a computing engine (the FSA) to real computers. Every time you step on the in/out mat for a door at a store, you are implementing a FSA. (Note I bow to your "door" above :)

Call it "Applied Mathematics" if you'd prefer. But it certainly has a very high reality coefficient. There is no ambiguity and there is semantic binding.

(Note: I realize that ABM does deal with this, and we've dealt with it with your MOTH model, but it is not necessarily general.)

Let me simplify. Is there a realm in which philosophy can exhibit a bug? And more specifically by simply "running" the philosophy engine?

I believe this may be possible, but I'm not sure. Maybe we'd have to create a new field. Certainly Turing, Church, von Neumann, Shannon, and many other in the computational world did. They stood on a brink, vital for going forward. Von Neumann had to argue for a computer to be admitted to the Institute for Advanced Study in Princeton .. it was considered just a machine. Church and Turing showed that to be nonsense. Can we do the same for philosophy?

NB: I'm not referring to "computational complexity" in which we deal with the running time issues of an algorithm, but to the semantics of computation itself. We really do have a strong grasp on what computation is and we do not quibble about meaning .. at least without heading immediately to axiomatic solutions.

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com

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