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