The Incomputable. Classifying the Theories of Physics. Sponsored by Templeton Foundation. June 21,

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1 Ger & Har Sci Pop O Laws Hierarchy The Incomputable Classifying the Theories of Physics 1,2,3 Sponsored by Templeton Foundation 1 Departamento de Matemática, Instituto Superior Técnico fgc@math.ist.utl.pt 2 CMAF Centro de Matemática e Aplicações Fundamentais 3 CFCUL Centro de Filosofia das Ciências da Universidade de Lisboa June 21, 2012

2 Ger & Har Sci Pop O Laws Hierarchy Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

3 Ger & Har Sci Pop O Laws Hierarchy Introduction: Newton-Bentley s correspondence Newton-Bentley s correspondence led Newton to abandon the Stoic Cosmos of a finite distribution of matter in infinite space and to adopt the Atomist Universe in which matter is distributed throughout infinite space.

4 Ger & Har Sci Pop O Laws Hierarchy Introduction: Newton-Bentley s correspondence Newton-Bentley s correspondence led Newton to abandon the Stoic Cosmos of a finite distribution of matter in infinite space and to adopt the Atomist Universe in which matter is distributed throughout infinite space.

5 Ger & Har Sci Pop O Laws Hierarchy Letter 1 If the distribution of matter were finite, then the matter on the outside of this space would by its gravity tend toward the matter on the inside, and by consequence, fall down into the middle of the whole space, and there compose one great spherical mass... But if the matter was evenly diffused through an infinite space, it would never convene into one mass but some of it into one mass and some of it into another so as to make an infinite number of great masses scattered at great distances from one to another throughout all of infinite space. And thus might the Sun and fixed stars be formed.

6 Ger & Har Sci Pop O Laws Hierarchy Letter 1 If the distribution of matter were finite, then the matter on the outside of this space would by its gravity tend toward the matter on the inside, and by consequence, fall down into the middle of the whole space, and there compose one great spherical mass... But if the matter was evenly diffused through an infinite space, it would never convene into one mass but some of it into one mass and some of it into another so as to make an infinite number of great masses scattered at great distances from one to another throughout all of infinite space. And thus might the Sun and fixed stars be formed.

7 Ger & Har Sci Pop O Laws Hierarchy Letter 2 Newton had fully agreed with Bentley that gravity meant providence had created a universe of great precision. The hypothesis of deriving the frame of the world by mechanical principles from matter evenly spread through the heavens being inconsistent with my system, I had considered it very little before your letters put me upon it, and therefore trouble you with a line or two more about it...

8 Ger & Har Sci Pop O Laws Hierarchy Letter 2 Newton had fully agreed with Bentley that gravity meant providence had created a universe of great precision. The hypothesis of deriving the frame of the world by mechanical principles from matter evenly spread through the heavens being inconsistent with my system, I had considered it very little before your letters put me upon it, and therefore trouble you with a line or two more about it...

9 Ger & Har Sci Pop O Laws Hierarchy Letter 2 Newton had fully agreed with Bentley that gravity meant providence had created a universe of great precision. The hypothesis of deriving the frame of the world by mechanical principles from matter evenly spread through the heavens being inconsistent with my system, I had considered it very little before your letters put me upon it, and therefore trouble you with a line or two more about it...

10 Ger & Har Sci Pop O Laws Hierarchy Letter 3 Newton elaborated earlier arguments that a divine power was essential in the design of initial conditions.... this frame of things could not always subsist without a divine power to conserve it.

11 Ger & Har Sci Pop O Laws Hierarchy Letter 3 Newton elaborated earlier arguments that a divine power was essential in the design of initial conditions.... this frame of things could not always subsist without a divine power to conserve it.

12 Ger & Har Sci Pop O Laws Hierarchy Letter 3 Newton elaborated earlier arguments that a divine power was essential in the design of initial conditions.... this frame of things could not always subsist without a divine power to conserve it.

13 Ger & Har Sci Pop O Laws Hierarchy Letter 3 Figure: Sensorium Dei in Newton s metaphysics.

14 Ger & Har Sci Pop O Laws Hierarchy Introduction: Newton s system of the world Newton, equipped with his system of the world, was facing incomputability in Nature...

15 Ger & Har Sci Pop O Laws Hierarchy Introduction: Newton s system of the world Newton, equipped with his system of the world, was facing incomputability in Nature...

16 Ger & Har Sci Pop O Laws Hierarchy Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

17 Ger & Har Sci Pop O Laws Hierarchy The Incomputable in Physics

18 Ger & Har Sci Pop O Laws Hierarchy Geroch and Hartle I [GH86] Regard number w as measurable if there exists a finite set of instructions for performing an experiment such that a technician, given an abundance of unprepared raw materials and an allowed error ε, is able by following those instructions to perform the experiment, yielding ultimately a rational number within ε of w.

19 Ger & Har Sci Pop O Laws Hierarchy Geroch and Hartle I [GH86] Regard number w as measurable if there exists a finite set of instructions for performing an experiment such that a technician, given an abundance of unprepared raw materials and an allowed error ε, is able by following those instructions to perform the experiment, yielding ultimately a rational number within ε of w.

20 Ger & Har Sci Pop O Laws Hierarchy Geroch and Hartle II [GH86] Every computable number is measurable... We now ask whether, conversely, every measurable number is computable or, in more detail, whether current physical theories are such that their measurable numbers are computable. This question must be asked with care.

21 Ger & Har Sci Pop O Laws Hierarchy Geroch and Hartle II [GH86] Every computable number is measurable... We now ask whether, conversely, every measurable number is computable or, in more detail, whether current physical theories are such that their measurable numbers are computable. This question must be asked with care.

22 Ger & Har Sci Pop O Laws Hierarchy Geroch and Hartle II [GH86] Every computable number is measurable... We now ask whether, conversely, every measurable number is computable or, in more detail, whether current physical theories are such that their measurable numbers are computable. This question must be asked with care.

23 Ger & Har Sci Pop O Laws Hierarchy Barry Cooper [Coo12] So there is a qualitatively different apparent breakdown in computability of natural laws at the quantum level the measurement problem challenges us to explain how certain quantum mechanical probabilities are converted into a well-defined outcome following a measurement. In the absence of a plausible explanation, one is denied a computable prediction. The physical significance of the Turing model depends upon its capacity for explaining what is happening here. If the phenomenon is not composite, it does need to be related in a clear way to a Turing universe designed to model computable causal structure. We look more closely at definability and invariance.

24 Ger & Har Sci Pop O Laws Hierarchy Barry Cooper [Coo12] So there is a qualitatively different apparent breakdown in computability of natural laws at the quantum level the measurement problem challenges us to explain how certain quantum mechanical probabilities are converted into a well-defined outcome following a measurement. In the absence of a plausible explanation, one is denied a computable prediction. The physical significance of the Turing model depends upon its capacity for explaining what is happening here. If the phenomenon is not composite, it does need to be related in a clear way to a Turing universe designed to model computable causal structure. We look more closely at definability and invariance.

25 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

26 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

27 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

28 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

29 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

30 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

31 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

32 Ger & Har Sci Pop O Laws Hierarchy Purpose 1 Barry Cooper has been emphasising that the Turing world can model the incomputability of the physical world... 2 Might well be that degree theory does not easily establish a clear link between Computability and Physics... 3 On the other side, the recursion theoretic model of identification has been developed with some connections with Physics... 4 However, their mentors do not accept that expected reality is possibly non-computable... 5 On yet another side, Gregory Chaitin has been claiming that a theory of Physics can be seen as a compression of (potential infinite) physical observations... 6 In this work, we will try to unite all these different, but confluent views of the world.

33 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

34 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification The Scientist Concept

35 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Boyle s law Example (Scientist Boyle) Boyle s law : The pressure of an ideal gas inside a flexible container, maintained at a constant temperature during a process of expansion or contraction, is proportional the the inverse of its volume. pv = const The scientist Boyle, on inputting text like this 5, , , , outputs the code e for the instance of Boyle s law.

36 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Boyle s law Example (Scientist Boyle) Boyle s law : The pressure of an ideal gas inside a flexible container, maintained at a constant temperature during a process of expansion or contraction, is proportional the the inverse of its volume. pv = const The scientist Boyle, on inputting text like this 5, , , , outputs the code e for the instance of Boyle s law.

37 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Boyle s law Example (Scientist Boyle) Boyle s law : The pressure of an ideal gas inside a flexible container, maintained at a constant temperature during a process of expansion or contraction, is proportional the the inverse of its volume. pv = const The scientist Boyle, on inputting text like this 5, , , , outputs the code e for the instance of Boyle s law.

38 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Boyle s law Example (Scientist Boyle) Boyle s law : The pressure of an ideal gas inside a flexible container, maintained at a constant temperature during a process of expansion or contraction, is proportional the the inverse of its volume. pv = const The scientist Boyle, on inputting text like this 5, , , , outputs the code e for the instance of Boyle s law.

39 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientists work with text! Text et al. 1 A text T for a function is a map of type N N N. By T [t] we denote the sequence of the first t elements of T. 2 INIT = {T [t] : T is a text for a function and t N}. Definition (Scientist, Gold [Gol67]) A scientist is a function (possibly partial, not necessarily computable) of type INIT N.

40 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientists work with text! Text et al. 1 A text T for a function is a map of type N N N. By T [t] we denote the sequence of the first t elements of T. 2 INIT = {T [t] : T is a text for a function and t N}. Definition (Scientist, Gold [Gol67]) A scientist is a function (possibly partial, not necessarily computable) of type INIT N.

41 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientists work with text! Text et al. 1 A text T for a function is a map of type N N N. By T [t] we denote the sequence of the first t elements of T. 2 INIT = {T [t] : T is a text for a function and t N}. Definition (Scientist, Gold [Gol67]) A scientist is a function (possibly partial, not necessarily computable) of type INIT N.

42 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientists work with text! Text et al. 1 A text T for a function is a map of type N N N. By T [t] we denote the sequence of the first t elements of T. 2 INIT = {T [t] : T is a text for a function and t N}. Definition (Scientist, Gold [Gol67]) A scientist is a function (possibly partial, not necessarily computable) of type INIT N.

43 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientists work with text! Text et al. 1 A text T for a function is a map of type N N N. By T [t] we denote the sequence of the first t elements of T. 2 INIT = {T [t] : T is a text for a function and t N}. Definition (Scientist, Gold [Gol67]) A scientist is a function (possibly partial, not necessarily computable) of type INIT N.

44 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Depicting a scientist General Identification SCIENTIST M From some order p N on... e 0#2#4#6#... Output stabilizes φ e(x) = 2x Figure: For all t p, scientist M on input ψ[t] outputs code e of ψ.

45 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Depicting a scientist General Identification SCIENTIST M From some order p N on... e 0#2#4#6#... Output stabilizes φ e(x) = 2x Figure: For all t p, scientist M on input ψ[t] outputs code e of ψ.

46 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Depicting a scientist General Identification SCIENTIST M From some order p N on... e 0#2#4#6#... Output stabilizes φ e(x) = 2x Figure: For all t p, scientist M on input ψ[t] outputs code e of ψ.

47 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Depicting a scientist General Identification SCIENTIST M From some order p N on... e 0#2#4#6#... Output stabilizes φ e(x) = 2x Figure: For all t p, scientist M on input ψ[t] outputs code e of ψ.

48 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Depicting a scientist General Identification SCIENTIST M From some order p N on... e 0#2#4#6#... Output stabilizes φ e(x) = 2x Figure: For all t p, scientist M on input ψ[t] outputs code e of ψ.

49 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Success for functions Definition (Scientific success on a single function, Gold [Gol67]) Let ψ : N N a total function. We say that scientist M identifies ψ if there exists an e N and an order p such that, for t p, M(ψ[t]) = e and φ e = ψ. Definition (... on a collection of functions, Gold [Gol67]) Let S be a set of total functions. We say that scientist M identifies S just in case she identifies every ψ S.

50 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Success for functions Definition (Scientific success on a single function, Gold [Gol67]) Let ψ : N N a total function. We say that scientist M identifies ψ if there exists an e N and an order p such that, for t p, M(ψ[t]) = e and φ e = ψ. Definition (... on a collection of functions, Gold [Gol67]) Let S be a set of total functions. We say that scientist M identifies S just in case she identifies every ψ S.

51 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Success for functions Definition (Scientific success on a single function, Gold [Gol67]) Let ψ : N N a total function. We say that scientist M identifies ψ if there exists an e N and an order p such that, for t p, M(ψ[t]) = e and φ e = ψ. Definition (... on a collection of functions, Gold [Gol67]) Let S be a set of total functions. We say that scientist M identifies S just in case she identifies every ψ S.

52 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Boyle EX -identification SCIENTIST From some order p N on... e 5, , , Output stabilizes Boyle φ e is an instance of Boyle s law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Boyle s law for the particular ideal gas under consideration.

53 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Boyle EX -identification SCIENTIST From some order p N on... e 5, , , Output stabilizes Boyle φ e is an instance of Boyle s law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Boyle s law for the particular ideal gas under consideration.

54 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Boyle EX -identification SCIENTIST From some order p N on... e 5, , , Output stabilizes Boyle φ e is an instance of Boyle s law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Boyle s law for the particular ideal gas under consideration.

55 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Boyle EX -identification SCIENTIST From some order p N on... e 5, , , Output stabilizes Boyle φ e is an instance of Boyle s law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Boyle s law for the particular ideal gas under consideration.

56 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Boyle EX -identification SCIENTIST From some order p N on... e 5, , , Output stabilizes Boyle φ e is an instance of Boyle s law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Boyle s law for the particular ideal gas under consideration.

57 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Popper on precision in [Pop35] Assume that the consequences of two theories differ so little in all fields of application that the very small differences between the calculated observable events cannot be detected, owing to the fact that the degree of precision attainable in our measurements is not sufficiently high. It will then be impossible to decide by experiments between the two theories, without first improving our technique of measurements. This shows that the prevailing technique of measurement determines a certain range a region within which discrepancies between observations are permitted by the theory.

58 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Popper on precision in [Pop35] Assume that the consequences of two theories differ so little in all fields of application that the very small differences between the calculated observable events cannot be detected, owing to the fact that the degree of precision attainable in our measurements is not sufficiently high. It will then be impossible to decide by experiments between the two theories, without first improving our technique of measurements. This shows that the prevailing technique of measurement determines a certain range a region within which discrepancies between observations are permitted by the theory.

59 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Van der Walls EX -identification SCIENTIST From some order p N on... e p 0, V 0 p 1, V 1... p k, V k... Output stabilizes Van der Walls φ e is an instance of Van der Walls law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Van der Walls law for the particular gas under consideration: (p + an2 V 2 )(V nb) = const.

60 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Van der Walls EX -identification SCIENTIST From some order p N on... e p 0, V 0 p 1, V 1... p k, V k... Output stabilizes Van der Walls φ e is an instance of Van der Walls law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Van der Walls law for the particular gas under consideration: (p + an2 V 2 )(V nb) = const.

61 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Van der Walls EX -identification SCIENTIST From some order p N on... e p 0, V 0 p 1, V 1... p k, V k... Output stabilizes Van der Walls φ e is an instance of Van der Walls law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Van der Walls law for the particular gas under consideration: (p + an2 V 2 )(V nb) = const.

62 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Van der Walls EX -identification SCIENTIST From some order p N on... e p 0, V 0 p 1, V 1... p k, V k... Output stabilizes Van der Walls φ e is an instance of Van der Walls law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Van der Walls law for the particular gas under consideration: (p + an2 V 2 )(V nb) = const.

63 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Scientist Van der Walls EX -identification SCIENTIST From some order p N on... e p 0, V 0 p 1, V 1... p k, V k... Output stabilizes Van der Walls φ e is an instance of Van der Walls law Figure: For all t p, scientist M on input Volume[Pressure] outputs the instance of Van der Walls law for the particular gas under consideration: (p + an2 V 2 )(V nb) = const.

64 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification EXplain and Behavioural Correct classes Definition (EX n -identification, Gold [Gol67], Blum and Blum [BB75], Case and Smith [CS78, CS83]) A set S of (total) recursive functions is said to belong to class EX n, if there exists a scientist M such that, for each ψ S, there exists an order p N such that, for all t p, M on input ψ[t] outputs the same n-variant code for ψ. Definition (BC n -identification, Bārzdiņ s [B 74], Feldman [Fel72], Case and Smith [CS78, CS83]) We say that a scientist M BC n -identifies a function ψ R, if there exists an order p N such that, for all t p, φ M(ψ[t]) is a n-variant code for ψ. We say that scientist M BC n -identifies a set of functions S R just in case she BC n -identifies every ψ S.

65 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification EXplain and Behavioural Correct classes Definition (EX n -identification, Gold [Gol67], Blum and Blum [BB75], Case and Smith [CS78, CS83]) A set S of (total) recursive functions is said to belong to class EX n, if there exists a scientist M such that, for each ψ S, there exists an order p N such that, for all t p, M on input ψ[t] outputs the same n-variant code for ψ. Definition (BC n -identification, Bārzdiņ s [B 74], Feldman [Fel72], Case and Smith [CS78, CS83]) We say that a scientist M BC n -identifies a function ψ R, if there exists an order p N such that, for all t p, φ M(ψ[t]) is a n-variant code for ψ. We say that scientist M BC n -identifies a set of functions S R just in case she BC n -identifies every ψ S.

66 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification EXplain and Behavioural Correct classes Definition (EX n -identification, Gold [Gol67], Blum and Blum [BB75], Case and Smith [CS78, CS83]) A set S of (total) recursive functions is said to belong to class EX n, if there exists a scientist M such that, for each ψ S, there exists an order p N such that, for all t p, M on input ψ[t] outputs the same n-variant code for ψ. Definition (BC n -identification, Bārzdiņ s [B 74], Feldman [Fel72], Case and Smith [CS78, CS83]) We say that a scientist M BC n -identifies a function ψ R, if there exists an order p N such that, for all t p, φ M(ψ[t]) is a n-variant code for ψ. We say that scientist M BC n -identifies a set of functions S R just in case she BC n -identifies every ψ S.

67 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification EXplain and Behavioural Correct classes Definition (EX n -identification, Gold [Gol67], Blum and Blum [BB75], Case and Smith [CS78, CS83]) A set S of (total) recursive functions is said to belong to class EX n, if there exists a scientist M such that, for each ψ S, there exists an order p N such that, for all t p, M on input ψ[t] outputs the same n-variant code for ψ. Definition (BC n -identification, Bārzdiņ s [B 74], Feldman [Fel72], Case and Smith [CS78, CS83]) We say that a scientist M BC n -identifies a function ψ R, if there exists an order p N such that, for all t p, φ M(ψ[t]) is a n-variant code for ψ. We say that scientist M BC n -identifies a set of functions S R just in case she BC n -identifies every ψ S.

68 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Almost and Everywhere Zero Example (AEZ) Standard example: The set AEZ of (total) computable functions ψ identical to 0 almost everywhere. Just consider the scientist M that, on input ψ(0)#... #ψ(t 1)#, builds the canonical list µ of the pairs t, ψ(t), where the ψ(t) is non-zero, and outputs the code of the programme λx.if x dom( µ), Then µ(x), Else 0. Such a scientist EX -identifies AEZ.

69 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Almost and Everywhere Zero Example (AEZ) Standard example: The set AEZ of (total) computable functions ψ identical to 0 almost everywhere. Just consider the scientist M that, on input ψ(0)#... #ψ(t 1)#, builds the canonical list µ of the pairs t, ψ(t), where the ψ(t) is non-zero, and outputs the code of the programme λx.if x dom( µ), Then µ(x), Else 0. Such a scientist EX -identifies AEZ.

70 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

71 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

72 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

73 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

74 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

75 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

76 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

77 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

78 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-collapsing hierarchy Proposition (The hierarchy of scientists, mainly from Case and Smith [CS78, CS83]) R / EX = EX 0 EX n EX BC R R / BC = BC 0 BC n BC R John Case writes in [Cas11]: Hence, tolerating anomalies strictly increases the inferring power as does relaxing the restriction of (syntactic) convergence to single programmes. Physicists use of slightly faulty explanations is vindicated!

79 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-union theorem Proposition (Blum and Blum [BB75], Case and Smith [CS78, CS83], Jain et al. [JORS99]) The classes EX n and EX are not closed under union. Theorem (Unification of scientific laws) Unification of scientific laws (as algorithms) is not always possible within the paradigm EX.

80 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-union theorem Proposition (Blum and Blum [BB75], Case and Smith [CS78, CS83], Jain et al. [JORS99]) The classes EX n and EX are not closed under union. Theorem (Unification of scientific laws) Unification of scientific laws (as algorithms) is not always possible within the paradigm EX.

81 Ger & Har Sci Pop O Laws Hierarchy Sci EX n and BC n Unification Non-union theorem Proposition (Blum and Blum [BB75], Case and Smith [CS78, CS83], Jain et al. [JORS99]) The classes EX n and EX are not closed under union. Theorem (Unification of scientific laws) Unification of scientific laws (as algorithms) is not always possible within the paradigm EX.

82 Ger & Har Sci Pop O Laws Hierarchy Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

83 Ger & Har Sci Pop O Laws Hierarchy Popper s Refutation Principle

84 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

85 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

86 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

87 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

88 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

89 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

90 Ger & Har Sci Pop O Laws Hierarchy Falsifiability [Popper states in the kernel of [Pop35]:] It has already been briefly indicated what rôle the basic statements play within the epistemological theory I advocate. We need them in order to decide whether a theory is to be called falsifiable, i.e. empirical [...] And we also need them for the corroboration of falsifying hypothesis, and thus for the falsification of theories [...] Basic statements must therefore satisfy the following conditions: (a) From a universal statement without initial conditions, no basic statement can be deduced. On the other hand, (b) a universal statement and a basic statement can contradict each other. Condition (b) can only be satisfied if it is possible to derive the negation of a basic statement from the theory which it contradicts. From this and condition (a) it follows that a basic statement must have a logical form such that its negation cannot be a basic statement in its turn.

91 Ger & Har Sci Pop O Laws Hierarchy Popper s refutability principle Popper s refutability principle, Case and Smith [CS78, CS83] The theory embedded in scientist M may not be refutable (!), for 1 It is not known if the instance φ e is undefined on some y; 2 Programme {e} on input y does not halt, i.e., one can not prepare any experimental apparatus to refute theory M on y, given a basic statement such as φ e (y) ψ(y), where φ e (y) is the prediction and ψ(y) is the observation, since it is not even known with generality if {e}(y) halts or not and, consequently, produce a prediction refutable by observation.

92 Ger & Har Sci Pop O Laws Hierarchy Popper s refutability principle Popper s refutability principle, Case and Smith [CS78, CS83] The theory embedded in scientist M may not be refutable (!), for 1 It is not known if the instance φ e is undefined on some y; 2 Programme {e} on input y does not halt, i.e., one can not prepare any experimental apparatus to refute theory M on y, given a basic statement such as φ e (y) ψ(y), where φ e (y) is the prediction and ψ(y) is the observation, since it is not even known with generality if {e}(y) halts or not and, consequently, produce a prediction refutable by observation.

93 Ger & Har Sci Pop O Laws Hierarchy Popper s refutability principle Popper s refutability principle, Case and Smith [CS78, CS83] The theory embedded in scientist M may not be refutable (!), for 1 It is not known if the instance φ e is undefined on some y; 2 Programme {e} on input y does not halt, i.e., one can not prepare any experimental apparatus to refute theory M on y, given a basic statement such as φ e (y) ψ(y), where φ e (y) is the prediction and ψ(y) is the observation, since it is not even known with generality if {e}(y) halts or not and, consequently, produce a prediction refutable by observation.

94 Ger & Har Sci Pop O Laws Hierarchy Popper s refutability principle Popper s refutability principle, Case and Smith [CS78, CS83] The theory embedded in scientist M may not be refutable (!), for 1 It is not known if the instance φ e is undefined on some y; 2 Programme {e} on input y does not halt, i.e., one can not prepare any experimental apparatus to refute theory M on y, given a basic statement such as φ e (y) ψ(y), where φ e (y) is the prediction and ψ(y) is the observation, since it is not even known with generality if {e}(y) halts or not and, consequently, produce a prediction refutable by observation.

95 Ger & Har Sci Pop O Laws Hierarchy Popper s refutability principle Popper s refutability principle, Case and Smith [CS78, CS83] The theory embedded in scientist M may not be refutable (!), for 1 It is not known if the instance φ e is undefined on some y; 2 Programme {e} on input y does not halt, i.e., one can not prepare any experimental apparatus to refute theory M on y, given a basic statement such as φ e (y) ψ(y), where φ e (y) is the prediction and ψ(y) is the observation, since it is not even known with generality if {e}(y) halts or not and, consequently, produce a prediction refutable by observation.

96 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

97 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Omniscient and Trivial Oracles

98 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Omniscient oracles Proposition (Adleman and Blum [AB91]) R O K EX.

99 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Omniscient oracles Definition (High sets) A set A is high if and only if K T A. Proposition (Adleman and Blum [AB91]) For all sets A, R O A EX if and only if A is high.

100 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Omniscient oracles Definition (High sets) A set A is high if and only if K T A. Proposition (Adleman and Blum [AB91]) For all sets A, R O A EX if and only if A is high.

101 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Omniscient oracles Definition (High sets) A set A is high if and only if K T A. Proposition (Adleman and Blum [AB91]) For all sets A, R O A EX if and only if A is high.

102 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Bounded queries Proposition If A T K, then O A, EX = EX. Proposition (Fortnow, Gasarch, Jain, Kinber, Kummer, Kurtz, Pleszkoch, Slaman, Solovay, and Stephan [FGJ + 94]) For all sets A, A T K if and only if, for all n N, O A,n EX O A,n+1 EX.

103 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Bounded queries Proposition If A T K, then O A, EX = EX. Proposition (Fortnow, Gasarch, Jain, Kinber, Kummer, Kurtz, Pleszkoch, Slaman, Solovay, and Stephan [FGJ + 94]) For all sets A, A T K if and only if, for all n N, O A,n EX O A,n+1 EX.

104 Ger & Har Sci Pop O Laws Hierarchy Omniscient Bounded Bounded queries Proposition If A T K, then O A, EX = EX. Proposition (Fortnow, Gasarch, Jain, Kinber, Kummer, Kurtz, Pleszkoch, Slaman, Solovay, and Stephan [FGJ + 94]) For all sets A, A T K if and only if, for all n N, O A,n EX O A,n+1 EX.

105 Ger & Har Sci Pop O Laws Hierarchy Scale Ordinal scales PRED π Degrees Overview 1 The Incomputable in Physics 2 The Scientist concept The Scientist concept EXplain and Behavioural Correct classes Unification of theories 3 Popper s Refutation Principle 4 Omniscient and Trivial Oracles Omniscient oracles Bounded queries 5 Identifying the Non-computable Scale functions Hierarchies of scales PRED π generalises BC Degree of regularity of a physical law 6 A New Non-collapsing Hierarchy Particular cases Another non-collapsing hierarchy Conclusion

106 Ger & Har Sci Pop O Laws Hierarchy Scale Ordinal scales PRED π Degrees Identifying the Non-computable Laws

107 Ger & Har Sci Pop O Laws Hierarchy Scale Ordinal scales PRED π Degrees Scale functions Definition (Scale function) We say that a (total) recursive function π : N N is a scale function if π is a non-decreasing surjective (linear or) sublinear function. Definition (Ordering scale functions) There is a relation between two total functions, ψ, ϕ : N N, by saying that ψ ϕ if ψ o(ϕ). This relation can be generalised to two classes of functions, F and G, by saying that F G if there exists a function ϕ G, such that for all functions ψ F, ψ ϕ.