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. N From: Friam [mailto:friam-boun...@redfish.com] 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 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 <nickthomp...@earthlink.net> 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-boun...@redfish.com] 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 <barry.mackic...@mackichan.com> 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 <o...@backspaces.net> wrote: On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson <nickthomp...@earthlink.net> 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
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