## Rational mechanics and natural mathematics (1995)

Venue: | In TAPSOFT'95 |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Pratt95rationalmechanics,

author = {Vaughan Pratt},

title = {Rational mechanics and natural mathematics},

booktitle = {In TAPSOFT'95},

year = {1995},

pages = {108--122},

publisher = {Springer}

}

### OpenURL

### Abstract

Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mechanics they cast Hamiltonian mechanics in terms of the interaction of body and mind. This paper addresses the chief stumbling block for Descartes ’ 17thcentury philosophy of mind-body dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of between-plane interaction alone. Lifting the two-valued Boolean logic of binary relations to the complex-valued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg-Schrödinger quantum-mechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physics [Pra94b]. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope of TAPSOFT. Bear in mind however that Boolean logic as the basis for

### Citations

240 | Modelling concurrency with partial orders - Pratt - 1986 |

227 | Semantical Considerations on Floyd-Hoare Logic - Pratt - 1976 |

88 | autonomous categories and linear logic - Barr - 1991 |

76 |
The Transactional Interpretation of Quantum Mechanics
- Cramer
- 1986
(Show Context)
Citation Context ...ace interactions more complex. These can be understood as having to go both backwards and forwards in time for their complete effect, the basis for Cramer’s transactional account of quantum mechanics =-=[Cra86]-=-, which Leslie Lamport drew to my attention in 1987. 5 Conclusion We have advanced a mechanism for the causal interaction of mind and body, and argued that separate additional mechanisms for body-body... |

60 | Residuated lattices - Dilworth, Ward - 1939 |

55 | Origins of the calculus of binary relations - Pratt - 1992 |

53 | Action logic and pure induction
- Pratt
- 1990
(Show Context)
Citation Context ...en this corresponds to the classical principle A → B ≡ ¬(A ∧ ¬B), as well as to linear logic’s A−◦B ≡ (A ⊗ B ⊥ ) ⊥ . It is a straightforward exercise to show the equivalence of these definitions; see =-=[Pra90a]-=- for further discussion. Definition (v) reveals the contravariance of the operation in R, and its covariance in T , composition being monotone in each argument, a form of bilinearity. 10sWe therefore ... |

49 |
Games semantics for linear logic
- Lafont, Streicher
- 1991
(Show Context)
Citation Context ...ly in the physical object A. States are possible, making a Chu space a kind of a Kripke structure [Gup93]: only one state at a time may be chosen from the menu X of alternatives. Lafont and Streicher =-=[LS91]-=- were the first to single out Chu spaces as a case of the more general Chu construction Chu(V, k) [Bar79, Bar91], namely V = Set, worthy of separate attention as a natural model of linear logic [Gir87... |

41 |
The Analysis of Matter
- Russell
- 1927
(Show Context)
Citation Context ...real substances. Berkeley opined that matter did not exist and that the universe consisted solely of ideas. Hobbes ventured 2sthe opposite: mind did not exist except as an artifact of matter. Russell =-=[Rus27]-=- embraced neutral monism, which reconciled Berkeley’s and Hobbes’ viewpoints as compatible dual accounts of a common neutral Leibnizian monad. This much of the history of mind-body dualism will suffic... |

38 | Dynamic algebras as a well-behaved fragment of relation algebras - Pratt - 1990 |

37 |
On the composition of processes
- Pratt
- 1982
(Show Context)
Citation Context ...precedence relation on events, namely a schedule of events, an alternative to automata theory and Kripke structures that has attracted our attention as a reliable model of true concurrency since 1982 =-=[Pra82]-=-. When we unravel the primitive causal links contributing to secondary causal interaction we find that two events, or two states, communicate with each other by interrogating all entities of the oppos... |

32 | Gages accept concurrent behavior
- Gupta, Pratt
- 1993
(Show Context)
Citation Context .... 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra =-=[GP93]-=-, metamathematics [Pra93, Pra94a], and physics [Pra94b]. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope o... |

29 | Temporal structures - Casley, Crew, et al. - 1991 |

26 | Varieties of relation algebras - Jónsson - 1982 |

22 | Event Spaces and Their Linear Logic
- Pratt
- 1991
(Show Context)
Citation Context ...natural transition relation on X. This is a partially ordered automaton. Elsewhere we have used higher dimensional automata to argue that automata could be reliably paired up as the dual of schedules =-=[Pra92]-=-. We find Chu spaces a very appealing extension of this duality. The left residual T/S, where T ⊆ U × W , S ⊆ V × W , is the dual of the right. We settle for defining T/S as the set of all pairs (u, v... |

12 |
Autonomous categories, LNM 752
- Barr
- 1979
(Show Context)
Citation Context ... interaction is of essentially the same form as the Heisenberg-Schrödinger quantum-mechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction =-=[Bar79]-=- strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physi... |

10 | Concurrent Kripke structures
- Gupta
- 1993
(Show Context)
Citation Context ...the dual Chu space A ⊥ = (X, A, |= ˘). Points have necessary existence, all being present simultaneously in the physical object A. States are possible, making a Chu space a kind of a Kripke structure =-=[Gup93]-=-: only one state at a time may be chosen from the menu X of alternatives. Lafont and Streicher [LS91] were the first to single out Chu spaces as a case of the more general Chu construction Chu(V, k) [... |

8 | Some constructions for order-theoretic models of concurrency - Pratt - 1985 |

8 | Chu spaces: Automata with quantum aspects
- Pratt
- 1994
(Show Context)
Citation Context ...rikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physics =-=[Pra94b]-=-. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope of TAPSOFT. Bear in mind however that Boolean logic as t... |

5 | An adjoint characterization of the category of sets
- Rosebrugh, Wood
- 1994
(Show Context)
Citation Context ...efore no mental plane to speak of in sets, making them the most physical of all the objects of traditional concrete (set-based) mathematics, if not of all category theory (and perhaps even there, cf. =-=[RW94]-=-). 1 Example: look for any vertex having exactly one edge to it from each vertex, and infinitely many edges out. There are lots of these in Set, namely the many singletons, all isomorphic, but none in... |

4 | spaces: Complementarity and uncertainty in rational mechanics - Chu - 1994 |

1 |
Descartes and Cartesianism
- Watson
- 1986
(Show Context)
Citation Context ...atible dual accounts of a common neutral Leibnizian monad. This much of the history of mind-body dualism will suffice as a convenient point of reference for the sequel. R. Watson’s Britannica article =-=[Wat86]-=- is a conveniently accessible starting point for further reading. The thesis of this paper is that mind-body dualism can be made to work via a theory that we greatly prefer to its monist competitors. ... |