(Paper Presented on 7th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, 4th World Congress and School on Universal Logic, 29th March 2013 – 7th April 2013, Rio de Janeiro, Brazil. See also the paper 42_Resolving_EPR_UNILOG_2015_Full_Paper and presentation on 26th June at the workshop on Emergent Computational Logics‘ at UNILOG’2015, 5th World Congress and School on Universal Logic, 20th June 2015 – 30th June 2015, Istanbul, Turkey.)

Some disturbing features of the standard Copenhagen interpretation of Quantum Theory

Amongst the philosophically disturbing features of the standard Copenhagen interpretation of Quantum Theory[1] are its:

$\bullet$ Essential indeterminateness [2];

$\bullet$ Essential separation of the world into system’ and observer’ [3].

$\bullet$ In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen noted [4] that accepting Quantum Theory, but denying these features of the Copenhagen interpretation, logically entails accepting either that the world is non-local [5] (thus contradicting Special Relativity), or that there are hidden variables [6] that would eliminate the need for accepting these features as necesary to any sound interpretation of Quantum Theory.

$\bullet$ In 1952 David Bohm proposed [7] an alternative mathematical development of the existing Quantum Theory (essentially equivalent to it, but based on Louis de Broglie’s pilot wave theory) whose interpretation appealed to hidden variables [8] (presumably hidden natural laws that were implicitly assumed by Bohm to be representable by well-defined classically computable mathematical functions [9]) that eliminated the need for indeterminism and the separation of the world into system’ and observer’.

$\bullet$ In 1964 John Stewart Bell showed [10] that any interpretation of Quantum Theory which appeals to (presumably classically computable) hidden variables in the above sense is necessarily non-local.

However, our foundational investigations into an apparently unrelated area [11] suggest that if the above presumption—concerning an appeal by Bohm and Bell to functions that are implicitly assumed to be classically computable—is correct, then the hidden variables in the Bohm-de Broglie interpretation of Quantum Theory could as well be presumed to involve natural laws that are mathematically representable only by functions that are algorithmically verifiable but not algorithmically computable [12] —in which case the interpretation might avoid being held as appealing to non-locality’.

The underlying perspective of this thesis

The underlying perspective of this thesis is that:

(1) Classical physics assumes that all the observable laws of nature can be mathematically represented in terms of well-defined functions that are algorithmically computable (as defined in the next section).

Since the functions are well-defined, their values are pre-existing and pre-determined as mappings that are capable of being known in their infinite totalities to an omniscient intelligence, such as Laplace’s intellect [13].

(2) However, the overwhelming experimental verification of the mathematical predictions of Quantum Theory suggests that the actual behavior of the real world cannot be assumed as pre-existing and pre-determined in this sense.

In other words, the consequences of some experimental interactions are theoretically incapable of being completely known in advance even to an omniscient intelligence, such as Laplace’s intellect.

So all the observable laws of nature cannot be represented mathematically in terms of functions that are algorithmically computable (as defined in the next section).

(3) Hence, either there is no way of representing all the observable laws of nature mathematically in a deterministic model, or all the observable laws of nature can be represented mathematically in a deterministic model—but in terms of functions that are computable’ in a non-predictable sense (we define these in the next section as algorithmically verifiable functions).

(4) The Copenhagen interpretation appears to opt for the first option in (3) above, and hold that there is no way of representing all the observable laws of nature mathematically in a deterministic model.

Hence the interpretation is not overly concerned with the seemingly essential non-locality of Quantum Theory, and its conflict with the deterministic mathematical representation of the laws of Special Relativity.

(5) The Bohm-de Broglie interpretation appears to reject the first option in (3) above, and to propose a way of representing all the observable laws of nature mathematically in terms of functions that are presumably taken implicitly to be algorithmically computable.

However, the Bohm-de Broglie interpretation has not so far been viewed as being capable of mathematically representing the seemingly essential non-local feature of Quantum Theory.

Let us therefore, for the moment, consider the second option in (3) above from the perspective suggested by the Birmingham paper; i.e., that the apparently non-local feature of Quantum Theory may actually be indicative of a non-constructive and counter intuitive-to-human-intelligence’ phenomena in nature that can, however, be mathematically represented by functions that are algorithmically verifiable, but not algorithmically computable.

Algorithmic verifiabilty and algorithmic computability

We define the two concepts [14]:

Algorithmic verifiability: A number-theoretical relation $F(x)$ is algorithmically verifiable if, and only if, for any given natural number $n$, there is an algorithm $AL_{(F,\ n)}$ which can provide objective evidence [15] for deciding the truth/falsity of each proposition in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

Algorithmic computability: A number theoretical relation $F(x)$ is algorithmically computable if, and only if, there is an algorithm $AL_{F}$ that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence $\{F(1), F(2), \ldots\}$.

We note that algorithmic computability implies the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions [16], whereas algorithmic verifiability does not imply the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

From the point of view of a finitary mathematical philosophy—which is the constraint within which an applied science ought to ideally operate—the significant difference between the two concepts could be expressed by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit [17]—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function.

We note that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

Theorem 1: There are mathematical functions that are algorithmically verifiable but not algorithmically computable.

Proof: (a) Since any real number is mathematically definable as the limit of a Cauchy sequence of rational numbers:

Let $R(n)$ denote the $n^{th}$ digit in the decimal expression of the real number $R$ in binary notation.

Then, for any given natural number $n$, there is an algorithm $AL_{(R,\ n)}$ that can decide the truth/falsity of each proposition in the finite sequence:

$\{R(1)=0, R(2)=0, \ldots, R(n)=0\}$.

Hence, for any real number $R$, the relation $R(x)=0$ is algorithmically verifiable trivially.

(b) Since it follows from Alan Turing’s Halting argument [18] that there are algorithmically uncomputable real numbers:

Let $[R(n)]$ denote the $n^{th}$ digit in the decimal expression of an algorithmically uncomputable real number $R$ in binary notation.

By (a), the relation $[R(x)=0]$ is algorithmically verifiable trivially.

However, by definition there is no algorithm $AL_{R}$ that can decide the truth/falsity of each proposition in the denumerable sequence:

$\{[R(1)=0], [R(2)=0], \ldots\}$.

Hence the relation $[R(x)=0]$ is algorithmically verifiable but not algorithmically computable. $\Box$

Some mathematical constants are definable only by algorithmically verifiable but not algorithmically computable functions

We note that:

(i) All the mathematically defined functions known to, and used by, science are algorithmically computable, including those that define transcendental numbers such as $\pi,\ e$, etc.

They can be computed algorithmically as they are all definable as the limit of some well-defined infinite series of rationals.

(ii) The existence of mathematical constants that are defined by functions which are algorithmically verifiable but not algorithmically computable—suggested most famously by Georg Cantor’s diagonal argument—has been a philosophically debatable deduction.

Such existential deductions have been viewed with both suspicion and scepticism by scientists such as Henri Poincaré, L. E. J. Brouwer, etc., and disputed most vociferously on philosophical grounds by Ludwig Wittgenstein [19].

(iii) A constructive definition of an arithmetical Boolean function $[R(x)]$ that is true (hence algorithmically verifiable [20]) but not provable in Peano Arithmetic (hence algorithmically not computable [21]) was given by Kurt Gödel in his 1931 paper on formally undecidable arithmetical propositions [22].

It was also disputed vociferously by Wittgenstein [23].

(iv) The definition of a number-theoretic function that is algorithmically verifiable but not algorithmically computable was also given by Alan Turing in his 1936 paper on computable numbers [24].

He defined a halting function, say $H(n)$, that is $0$ if, and only if, the Turing machine with code number $n$ halts on input $n$. Such a function is mathematically well-defined, but assuming that it defines an algorithmically computable real number leads to a contradiction.

Turing thereupon concluded the mathematical existence of algorithmically uncomputable real numbers.

(v) A definition of a number-theoretic function that is algorithmically verifiable but not algorithmically computable was given by Gregory Chaitin [25].

He defined a class of constants—denoted by $\Omega$—which is such that if $C(n)$ is the $n^{th}$ digit in the decimal expression of an $\Omega$ constant, then the function $C(x)$ is algorithmically verifiable but not algorithmically computable.

Some physical constants may be representable by real numbers that are definable only by algorithmically verifiable but not algorithmically computable functions

Now, we note that:

“… the numerical values of dimensionless physical constants are independent of the units used. These constants cannot be eliminated by any choice of a system of units. Such constants include:

$\bullet\ \alpha$, the fine structure constant, the coupling constant for the electromagnetic interaction ($\approx 1/137.036$). Also the square of the electron charge, expressed in Planck units. This defines the scale of charge of elementary particles with charge.

$\bullet\ \mu$ or $\beta$, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron ($\approx 1836.15$). More generally, the rest masses of all elementary particles relative to that of the electron.

$\bullet\ \alpha_{s}$, the coupling constant for the strong force ($\approx 1$)

$\bullet\ \alpha{G}$, the gravitational coupling constant ($\approx 10^{-38}$) which is the square of the electron mass, expressed in Planck units. This defines the scale of the mass of elementary particles.

At the present time, the values of the dimensionless physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics. …

The list of fundamental dimensionless constants decreases when advances in physics show how some previously known constant can be computed in terms of others. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. A successful Theory of Everything’ would allow such a calculation, but so far, this goal has remained elusive.”

This suggests that:

Thesis 1: Some of the dimensionless physical constants are only representable in a mathematical language as real numbers that are defined by functions which are algorithmically verifiable, but not algorithmically computable.

In other words, we cannot treat such constants as denoting—even in principle—a measurable limit, as we could a constant that is representable mathematically by a real number that is definable by algorithmically computable functions.

From the point of view of mathematical philosophy, this distinction would be expressed by saying that the sequence of digits in the decimal representation of an unmeasurable’ physical constant cannot be treated in a mathematical language as a completed’ infinite sequence, whilst the corresponding sequence in the decimal representation of a measurable’ physical constant can be treated as a completed’ infinite sequence.

Zeno’s argument: The dichotomy between a continuous physical reality and its discretely representable mathematical theory

Zeno’s paradoxical arguments highlight the philosophical and theological dichotomy between our essentially continuous’ perception of the physical reality that we seek to capture with our measurements, and the essential discreteness’ of any mathematical language of Arithmetic in which we seek to express such measurements.

The distinction between algorithmic verifiability and algorithmic computability of Arithmetical functions can be seen as reflecting the dichotomy mathematically.

Classical laws of nature

For instance, the distinction suggests that classical mechanics can be held as complete with respect to the algorithmically computable representation of the physical world, in the sense that:

Thesis 2: Classical laws of nature determine the nature and behaviour of all those properties of the physical world which are mathematically describable completely at any moment of time $t(n)$ by algorithmically computable functions from a given initial state at time $t(0)$.

Neo-classical laws of nature

On the other hand, the distinction also suggests that:

Thesis 3: Neo-classical laws of nature determine the nature and behaviour of those properties of the physical world which are describable completely at any moment of time $t(n)$ by algorithmically verifiable functions; however such properties are not completely describable by algorithmically computable functions from any given initial state at time $t(0)$.

Since such behaviour [26] follows fixed laws and is determinate (even if not algorithmically predictable by classical laws), Albert Einstein could have been justified in his belief that:

“… God doesn’t play dice with the world”

and in holding that:

“I like to think that the moon is there even if I am not looking at it”.

Incompleteness: Arithmetical analogy

The distinction also suggests that neither classical mechanics nor neo-classical quantum mechanics can be described as mathematically complete’ with respect to the algorithmically verifiable behaviour of the physical world.

The analogy here is that Gödel showed in 1931 [27] that any formal arithmetic is not mathematically complete with respect to the algorithmically verifiable nature and behaviour of the natural numbers [28].

However it can be argued that the first-order Peano Arithmetic PA is complete [29] with respect to the algorithmically computable nature and behaviour of the natural numbers.

In this sense, the $EPR$ paper may not be entirely wrong in holding that:

“We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.”

Conjugate properties

The above also suggests that:

Thesis 4: The nature and behaviour of two conjugate properties $F_{1}$ and $F_{2}$ of a particle $P$ that are determined by neo-classical laws are described mathematically at any time $t(n)$ by two algorithmically verifiable, but not algorithmically computable, functions $f_{1}$ and $f_{2}$.

In other words, it is the very essence of the neo-classical laws determining the nature and behaviour of the particle that—at any time $t(n)$—we can only determine either $f_{1}(n)$ or $f_{2}(n)$, but not both.

Hence measuring either one makes the other indeterminate as we cannot go back in time.

However, this does not contradict the assumption that any property of an object must obey some deterministic natural law for any possible measurement that is made at any time.

Entangled particles

The above similarly suggests that:

Thesis 5: The nature and behaviour of an entangled property of two particles $P$ and $Q$ is determined by neo-classical laws and are describable mathematically at any time $t(n)$ by two algorithmically verifiable—but not algorithmically computable—functions $f_{1}$ and $g_{1}$.

In other words, it is the very essence of the neo-classical laws determining the nature and behaviour of the entangled properties of two particles that—at any time $t(n)$—determining one immediately gives the state of the other without measurement since the properties are entangled.

Again, this does not contradict the assumption that any property of an object must obey some deterministic natural law for any possible measurement that is made at any time.

Nor does it require any information to travel from one particle to another consequent to a measurement.[29a]

Schrödinger’s cat

If $[F(x)]$ is an algorithmically verifiable but not algorithmically computable Boolean function, we can take the query:

Is $F(n)=0$ for all natural numbers?

as corresponding mathematically to the Schrödinger question:

Is the cat dead or alive at any given time $t$?

We can then argue that there is no mathematical paradox involved in Schrödinger‘s assertion that the cat is both dead and alive, if we take this to mean that:

I may either assume the cat to be alive until a given time $t$ (in the future), or assume the cat to be dead until the time $t$, without arriving at any logical contradiction in my existing Quantum description of nature.

In other words:

Once we accept Quantum Theory as a valid description of nature, then there is no paradox in stating that the theory essentially cannot predict the state of the cat at any moment of future time (so the inability to predict does not arise out of a lack of sufficient information about the laws of the system that Quantum theory is describing, but stems from the very nature of these laws).

The mathematical analogy for the above would be:

Once we accept that Peano Arithmetic is consistent [30] and categorical [31] then we cannot deduce from the axioms of PA whether $F(n)=0$ for all natural numbers, or whether $F(n)=1$ for some natural number.

Conclusion
To sum up, we suggest that the paradoxical element in the $EPR$ argument may disappear if we could argue that:

(i) All properties of physical reality can be deterministic—in the sense that any physical property can have one, and only one, value at any time $t(n)$—where the value is completely determined by some natural law which need not, however, be algorithmic.

(ii) There are elements of such a physical reality whose properties at any time $t(n)$ are determined completely in terms of their putative properties at some earlier time $t(0)$.

Such properties are predictable mathematically since they are representable mathematically by algorithmically computable functions.

The Laws of Classical Mechanics describe the nature and behaviour of such physical reality only.

(iii) There can be elements of such a physical reality whose properties at any time $t(n)$ cannot be theoretically determined completely from their putative properties at some earlier time $t(0)$.

Such properties are unpredictable mathematically since they are only representable mathematically by algorithmically verifiable, but not algorithmically computable, functions.

The Laws of Quantum Mechanics describe the nature and behaviour of such physical reality.

The need for constructive mathematical foundations

The following perspective [32] emphasises the need for a universally common, constructive, foundation for the mathematical representation of elements of reality such as those considered above:

“Our investigations lead us to consider the possibilities for reuniting the antipodes’. The antipodes being classical mathematics (CLASS) and intuitionism (INT).

… It therefore seems worthwhile to explore the formal’ common ground of classical and intuitionistic mathematics. If systematically developed, many intuitionistic results would be seen to hold classically as well, and thus offer a way to develop a strong constructive theory which is still consistent with the rest of classical mathematics.

Such a constructive theory can form a conceptual framework for applied mathematics and information technology.

These sciences now use an ad-hoc approach to reality since the classical framework is inadequate. … and can easily use the richness of ideas already present in classical mathematics, if classical mathematics were to be systematically developed along the common grounds before the unconstructive elements are brought in.”

Frank Waaldijk [33]

“… we propose that Laplacian determinism be seen in the light of constructive mathematics and Church’s Thesis.

This means amongst other things that infinite sequences (of natural numbers; a real number is then given by such an infinite sequence) are never finished’, instead we see them developing in the course of time.

Now a very consequent, therefore elegant interpretation of Laplacian determinism runs as follows.

Suppose that there is in the real world a developing-infinite sequence of natural numbers, say $\alpha$. Then how to interpret the statement that this sequence is uniquely determined’ by the state of the world at time zero?

At time zero we can have at most finite information since, according to our constructive viewpoint, infinity is never attained. So this finite information about $\alpha$ supposedly enables us to uniquely determine’ $\alpha$ in its course of time.

It is now hard to see another interpretation of this last statement, than the one given by Church’s Thesis, namely that this finite information must be a (Turing-)algorithm that we can use to compute $\alpha (n)$ for any $n \in \mathbb(N)$.

With classical logic and omniscience, the previous can be stated thus:

for every (potentially infinite) sequence of numbers $(a_{n})_{n \in \mathbb{N}}$ taken from reality there is a recursive algorithm $\alpha$ such that $\alpha (n) = a_{n}$ for each $n \in \mathbb{N}$. This statement is sometimes denoted as CT$_{phys}$‘,

… this classical omniscient interpretation is easily seen to fail in real life.

Therefore we adopt the constructive viewpoint.

The statement the real world is deterministic’ can then best be interpreted as:

a (potentially infinite) sequence of numbers $(a_{n})_{n \in \mathbb{N}}$ taken from reality cannot be apart from every recursive algorithm $\alpha$ (in symbols: $\neg \forall \alpha \in \sigma_{\omega REC} \exists n \in \mathbb{N}\ [\alpha(n) \neq a_{n}]$)’.”

Frank Waaldijk [34]

References

Be64 John Stewart Bell. 1964. On the Einstein Podolsky Rosen Paradox Physics Vol 1, No. 3, pp.195-200, 1964. Reprinted in J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge, 2004, p. 14-21.

Bo52 David Bohm. 1952. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables’, I. Physical Review 85 n.2, pp.166-179 (1952) and A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables’, II. Physical Review 85 n.2, pp.180-193 (1952).

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic. (4th ed). Cambridge University Press, Cambridge.

Ch82 Gregory J. Chaitin. 1982. Gödel’s Theorem and Information. International Journal of Theoretical Physics 22 (1982), pp. 941-954.

EPR35 Albert Einstein, Boris Yakovlevich Podolsky, Nathan Rosen. 1935. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? InPhysical Review PHYS REV X – vol. 47, no. 10, pp.777-780. Bibcode 1935PhRv\ldots 47…777E. doi:10.1103/PhysRev.47.777.

Go31 Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. pp.5-38.

Kl52 Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam.

Mu91 Chetan R. Murthy. 1991. An Evaluation Semantics for Classical Proofs. Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991.

Sc35 Erwin Schrödinger. Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaftern. 23: pp. 807-812; 823-823, 844-849. (1935). English translation: John D. Trimmer, The present situation in Quantum Mechanics, Proceedings of the American Philosophical Society 124, pp. 323-38 (1980), reprinted in Quantum Theory and Measurement, p. 152 (1983).

Sh11 Sheldon Goldstein et al. 2011. Bell’s Theorem, Scholarpedia, 6(10):8378, doi:10.4249/scholarpedia.8378.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

Wa03 Frank Waaldijk. 2003. On the foundations of constructive mathematics. Web paper.

Wi78 Ludwig Wittgenstein. 1937. Remarks on the Foundations of Mathematics. 1978 ed., MIT Press, Cambridge.

An12 Bhupinder Singh Anand. 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

An12a Bhupinder Singh Anand. 2012. Some consequences of interpreting the associated logic of the first-order Peano Arithmetic PA finitarily. Draft.

An13 … 2013. A suggested mathematical perspective for the EPR argument. Paper Presented on 7th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, 4th World Congress and School on Universal Logic, 29th March 2013 – 7th April 2013, Rio de Janeiro, Brazil.

Notes

Return to 1: “Because it consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th Century, there is no definitive statement of the Copenhagen interpretation”, Wikipedia; cf., Copenhagen Interpretation of Quantum Mechanics, Stanford Encyclopedia of Philosophy; also The Copenhagen Interpretation of Quantum Mechanics by Ben Best.

Return to 2: “It is a general principle of orthodox formulations of quantum theory that measurements of physical quantities do not simply reveal pre-existing or pre-determined values, the way they do in classical theories. Instead, the particular outcome of the measurement somehow “emerges” from the dynamical interaction of the system being measured with the measuring device, so that even someone who was omniscient about the states of the system and device prior to the interaction couldn’t have predicted in advance which outcome would be realized.” … Sh11.

Return to 2: Highlighted famously by Erwin Schrödinger’s caustic observation on the dubious condition of his Platonic pet:

“One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The $\psi$-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.” … Sc35, §5.

Return to 5: “Non-local’ … means that there exist interactions between events that are too far apart in space and too close together in time for the events to be connected even by signals moving at the speed of light.” … Sh11.

Return to 6: “Traditionally, the phrase hidden variables’ is used to characterize any elements supplementing the wave function of orthodox quantum theory.” … Sh11.

Return to 8: “This terminology is, however, particularly unfortunate in the case of the de Broglie-Bohm theory, where it is in the supplementary variables—definite particle positions—that one finds an image of the manifest world of ordinary experience.” … Sh11.

Return to 9: Which could be considered as having pre-existing or pre-determined mathematical values over the domain over which the functions are well-defined.

Return to 11: Concerning evidence-based and finitary interpretations of the first order Peano Arithmetic PA, in An12.

Return to 13: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain; and the future just like the past would be present before its eyes.” … Pierre Simon Laplace, A Philosophical Essay on Probabilities.

Return to 15: It is by now folklore … that one can view the values of a simple functional language as specifying evidence for propositions in a constructive logic …” … Mu91.

Return to 16: We note that the concept of algorithmic computability’ is essentially an expression of the more rigorously defined concept of realizability’ in Kl52, p.503.

Return to 17: In the sense of a physically completable’ infinite sequence (as needed to resolve Zeno’s paradox).

Return to 26: A putative model for such behaviour is given in Wa03, §1.5, p.5:

“The second way to model our real world is to assume that it is deterministic. …

It would be worthwhile to explore the consequences of a deterministic world with incomplete information (since under the assumption of determinancy in the author’s eyes this comes closest to real life).

That is a world in which each infinite sequence is given by an algorithm, which in most cases is completely unknown.

We can model such a world by introducing two players, where player I picks algorithms and hands out the computed values of these algorithms to player II, one at a time.

Sometimes player I discloses (partial) information about the algorithms themselves.

Player II can of course construct her or his own algorithms, but still is confronted with recursive elements of player I about which she/he has incomplete information.”

Return to 28: Which—as shown in An12—is the behaviour sought to be captured by the Standard interpretation of the first order Peano Arithmetic PA.

Return to 29a: For a model of widely separated outputs that are correlated but don’t allow communication, which demonstrates that non-locality doesn’t imply communication, see this paper Wim van Dam: Implausible Consequences of Superstrong Nonlocality and this explanatory blog A Neighborhood of Infinity: Distributed computing with alien technology.

Return to 31: An12a establishes the `categoricity’—which means that any two models of the Arithmetic are isomorphic.