The Dynamics of Formal Axiomatic Systems Nithin Nagaraj Mathematical Modelling Unit National Institute of Advanced Studies Email: [email protected] July 21, 2005

Abstract In this paper, we investigate whether Chaos theory (Non-Linear Dynamics) can be employed to study the behaviour of Formal Axiomatic Systems (FAS). Our preliminary experiments seems to suggest that there is an isomorphism between the logistic map and a purely typographical-FAS. Particularly, we demonstrate that a purely typographical FAS can be represented efficiently by a simple Tent-map. Similar to G¨odel’s arithematization of Typographical Number Theory (TNT), we claim that Dynamical Systems can be employed as an alternate representation of FAS by a suitable isomorphism. The tools, techniques and the rich vocabulary of Chaos Theory which have been highly efficient in studying non-linear physical phenomena may also be employed to study the Dynamics of FAS.

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Introduction

Meta-mathematics deals with the study of the nature of Mathematics by employing the tools of Logic (inference), Philosophy (analysis) and Mathematics itself. It primarily involves investigating the foundations of Mathematics, the basic Axioms and the rules of inference that determine the Truths or Theorems of Mathematics. How do we derive theorems from known axioms by efficiently applying the rules of inference ? The rules of inference, for example, in the case of Set Theory are the five axioms of Peano. The derivation of theorems is known as Proofs. The system of axioms together with the rules of inference is called a Formal Axiomatic System (FAS). Note that, since such an FAS contains in its essence, all the provable theorems of the system, it suffices to say that FAS is a collection 1

of axioms and rules of inference. One of the important topics of Mathematics has been the space of theorems as against non-theorems and to determine which of the theorems can be ‘reached’ by existing axioms (reachable truths). It is now well known, thanks to the celebrated Incompleteness Theorem of G¨odel [2], that for a sufficiently powerful axiomatic system, there exists unreachable truths. In other words, there exists theorems that can’t be proved to be True (and their negation as False). Other important questions pursued by researchers are − Does there exist shorter proofs for known theorems ?, What statements are independent of Number Theory (such as the Continuum Hypothesis and the Axiom of Choice) ? and other questions regarding the strength of FAS. In this paper, we investigate (what we define as) the Dynamics of Formal Axiomatic System. We are mainly interested in studying the behavior of FAS, as though it were a physical system. Of course, we barely scratch the surface of such a deep and profound topic. We begin with observing a purely typographical FAS (inspired by Douglas Hofstadter’s ‘G¨ odel, Escher, Bach’ [3]). We draw an isomorphism between an FAS and a Logistic Map under Chaos. We demonstrate that a simple Tent-map is an efficient representation of the purely typographical-FAS (the AB−system) under consideration. The dynamics of the Tent-map may throw more insight in to the dynamics of the FAS. The Logistic Map is an arsenal of a larger theory of Non-Linear Dynamics, popularly known as Chaos Theory [1]. Sensitive dependence to initial conditions is synonymous with Chaos (this is only a necessary condition, but not sufficient). The term Butterfly-effect (Gleick [4]) has been an oft-quoted example of a system under Chaos (the flutter of the wings of a butterfly in Texas can cause a tornado in Tokyo). Chaos was first observed in non-linear deterministic systems modelled for forecasting (predicting) long-term weather. It was observed that a slight change in the input parameters of the equations resulted in widely varying solutions. Today, it is well acknowledged that Chaos is everywhere [4] and that linear systems are more of an exception than the rule (as contrarily believed).

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Formal Axiomatic Systems (FAS)

Formal Axiomatic Systems (FAS) were invented by the American logician Emil Post in the 1920’s (known as Post Production System). Douglas Hofstadter, in his masterpiece, the pulitzer prize winning book ‘G¨ odel, Escher, Bach’ introduces FAS by means of a very interesting puzzle − the MU puzzle. He defines a typographical formal system − the M IU −system which consists of three letters M , I and U . This means that only strings which contain these letters are valid strings of the M IU −system. Further, he defines four rules to operate on strings to produce new strings (or theorems). However, there is only one starting string to begin with (axiom) and that is the string − M I. The specific question he poses is − whether the string M U can be formed by using the starting string (M I) and using the rules in any order and any number of times ?. In other words, for the M IU −system, is M U a theorem ?. He later argues that M U can *not* be a theorem and he arrives at this conclusion by a trick which was originally invented by G¨odel − arithmetization or G¨ odel numbering. He converts the purely typographical FAS (the M IU −system) into an arithmetical system (the 310−system) by a special isomorphism. Every typographical manipulation of the original M IU −system has an arithmetical equivalent in the 310−system. He then beautifully concludes that the number of 1s (or Is in the original system) in every valid theorem of the system can only be altered if the original string had a count of 3. Since 31 (M I) has count of 1 for 1 (I), it can in no way be reduced to zero and hance M U can never be derived from this axiom. Thus M U is not a theorem of the system (please refer to [3] for an illuminating discussion on the M IU system). What are the lessons one can learn from this unique isomorphism ?. Several:− 1. G¨odel numbering is an efficient technique for converting a typographical FAS in to an arithmetical one. 2. Typographical rules for manipulating strings are actually arithmetical rules for operating on numbers. 3. Just as any set of typographical rules generates a set of theorems, a corresponding set of natural numbers will be generated by repeated application of arithmetical rules. 3

4. Such generated natural numbers are called as producible numbers and their role in Number Theory is similar to that of theorems inside a FAS. Producible numbers are relative to a system of arithmetical rules. 5. Producible numbers form a recursively enumerable set, i.e., they can be produced by a recursive method (hence computable). 6. Finally, the most important aspect of this jugglery is that the transpose of a formal axiomatic system into number theory using such arithmetization is very useful in learning about the FAS. Number theory and it’s rich tools can be used to assail the properties of the isomorphic arithmetical system and thus throw insight in to the original FAS. Hence such isomorphisms are beneficial and could it be, therefore, that the means to answer any question about any formal system lies within just a single formal system − Number Theory. The aforementioned goal was the ambitious Hilbert’s program for formalizing all of Mathematics in to the finite axioms of Number Theory (particularly Typographical Number Theory or TNT) which was destroyed by G¨odel’s argument and his celebrated Incompleteness Theorem [2] which states that all consistent and sufficiently powerful axiomatic systems include undecidable propositions . However, that did not mean that nothing could be achieved by such arithmetization. Keeping in mind the undecidable G¨odelian statements, the string-space of a FAS is depicted in Figure 1.

2.1

Isomorphisms: Why are they important ?

Let us say a bit about isomorphisms. Isomorphisms are information preserving transformations. Isomorphism applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where ‘corresponding’ means that the two parts play similar roles in their respective structures. As we saw in the M IU −system, the isomorphism takes the typographical system in to an arithmetical one (actually invertible, hence it is always possible to come back to the original FAS). G¨odel numbering is one example of an useful isomorphism to study FAS by transforming it in to TNT. An important practical application of isomorphisms is the use of a G¨odel-type numbering in the field of Data Compression. Arithmetic Coding 4

FAS

"TRUE" Strings

"FALSE" Strings

Theorems

NonnonTheorems

Axioms

Un-Reachable Un-Reachable Truths Falsehoods Well-formed Strings All Possible Strings

Figure 1: The string-space of a Formal Axiomatic System (FAS).

[5], the popular data compression algorithm (an entropy coding method) treats every message as a real number in the range [0, 1]. The partitioning of this compact set is performed according to the probabilities of the letters of the alphabet of the message. Arithmetic coding comes arbitrarily close to the entropy of the message because it succeeds in allocating fractional bits to individual alphabets as against methods like Huffman Coding which give integer number of bits to the alphabets [5]. To put it in Douglas Hofstadter’s words - “It is a cause for joy when a mathematician discovers an isomorphism between two structures which he knows. It is often a “bolt from the blue” and a source of wonderment. The perception of an isomorphism between two known structures is a significant advance in knowledge − and I claim that it is such perceptions of isomorphism which create meanings in the minds of people...” There is in general, a look-out for new isomorphisms.

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The AB Typographical FAS

In this section, I shall describe a typographical FAS which has some interesting properties. Consider the alphabet Σ = {A, B} and all possible strings (the stringspace Σ∗ ) of A and B. The FAS system allows the following rules 5

for forming new strings from a starting string. A well-formed string of the FAS consists of only As and Bs, possibly an infinite number of them. • Rule 1: Any string of the form Ax can be transformed to x (x can be finite or an infinite string of Σ∗ ). • Rule 2: Any string of the form Bx can be subject to the following set of transformations in that order. First the string Bx is transformed to x. Next, an infinite string of As is appended to yield xA∞ (xAAA . . .). From this resulting string, As and Bs are interchanged to yield a new string y = xc B ∞ , where xc is the complementary string of x in the sense that all the As and Bs are interchanged. There are some additional rules for simplification. These are: • Rule 3: At any location of any string, an infinite number of As can be replaced by a single A. In other words, A∞ = A. • Rule 4: At any location of any string, AB ∞ can be replaced by B (AB ∞ = B). • Rule 5: The particular string BB ∞ = B ∞ = B. • Rule 6: For any string x, we can concatenate an infinite number of As at the end (x = xA∞ ). Now, we can start forming theorems of the FAS. But, I have not specified to you the starting strings or the axioms on which you can apply the aforementioned rules to derive new theorems or strings (for the sake of completeness, you can assume the NULL-string φ as a given axiom). This is where the fun starts. I would encourage you to play with the FAS with arbitrarily starting axioms. You will soon realize that the FAS has some very peculiar properties. To name a few: 1. If the string A is the only axiom, then we do not get any new theorems from the system. This is easy to see. First, observe that I can only apply the rules 1 and 6 in any order I choose to. No matter how I apply these rules, I end up in either the NULL-string or A which are already axioms. Hence no new theorems result by this choice. 2. Similarly, if I choose the string B to be the starting axiom, I do not get any new theorems. Convince yourself of this fact by applying rules 2 and as required rules 3 to 6. 6

3. What if I choose other strings as my initial axiom ? Say, for example, the string AB. I notice that I can derive theorem B by rule 1 and nothing more. Therefore, this FAS has only one theorem. 4. If I choose my axiom as BAB, then I am slightly lucky. I get two theorems namely BB and B. The question that would immediately come to one’s mind is − There are so many choices of initial axioms and in each of those cases, how many theorems would the FAS contain ?. This seems to be a very tough question to answer. We are also interested in determining whether a particular given string to us is a theorem of the FAS (with previously chosen axioms). If not, what axioms are to be chosen to ensure that the particular string turns out as a theorem ?. These questions are not arbitrary but are very relevant to the study of Mathematics. Notice, how these questions can be carried over to more complicated FAS of Mathematics. These play a crucial role in determining the power of a particular FAS. I am using the word ‘power’ in a very loose sense here. For instance, given two very closely related FAS, does there exist a computable procedure to determine which has more number of theorems with proofs. More theorems may not always mean more power, but seems to be one metric for determining the strength of a FAS. Finally, we are also interested in knowing whether the particular FAS is complete or incomplete. If it is incomplete, we would like to be able to construct statements which are undecidable in the FAS, just the thing that G¨odel proved for TNT.

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An Isomorphism − The Tent Map

Now, you might immediately retort and say that we can do the G¨odel numbering trick for the AB−FAS and thereby learn about the FAS. This may be very well possible, but we will not go down that path. We shall investigate a slightly different isomorphism. Although we do the same trick of arithmetization of the FAS, our interpretation of the arithmetical-FAS is not that of TNT, but one of belonging to Chaos Theory (one might argue that even Chaos Theory can be reduced to TNT, but we shall not do this because we wish to use the rich vocabulary of Chaos Theory along with it’s tools and techniques to analyze the FAS under consideration.) We perform a very simple isomorphism as follows. We treat all possible 7

Figure 2: Isomorphism of strings of the AB−system to real numbers of the closed set [0, 1].

strings of Σ∗ as binary expansions of real numbers in the range [0, 1] (see Figure 2). We make the substitution A = 0 and B = 1. We drop the conventional dot in these expansions for the sake of convenience. Notice how rules 3 and 4 correspond to the fact that in binary notation .101000 . . . = .101 and .0111 . . . = 0.1. This isomorphism maps the rules 1 and 2 to the following Tent map (Figure 3):

x 7→ 2x,

x≤

x 7→ 2(1 − x)

1 2 x>

1 2

So, we are now in the realm of Chaos Theory and we can start to apply it’s tools to analyze the FAS. Note that, by the above isomorphism, the axioms of the AB−system maps to the initial condition of the Tent map. We can also re-interpret the observations we made earlier on the number of theorems for different choice of the starting axioms. Since 0 (A∞ ) and 1 (B ∞ ) are fixed points of the Tent map (i.e. they get mapped to themselves under the above transformation), they do not produce any theorems. The initial condition of 0.5 gets iterated to 1 by the above Tent map. Similarly BAB corresponds to .101 in binary which is 0.625 in decimal. This gets iterated to 0.75 in decimal (0.11 in binary (BB)) and in the next iteration to 0.5 in decimal (0.1 in binary or B. From 0.5 it gets mapped to 1 and stays at 1. In our typographical system, we did not make the distinction between B and B ∞ , this only means that we treat 0.5 and 0.999 . . . (=1.0) as equivalent. Now, the set of questions which we posed in the earlier sections get translated to questions about the dynamical system which is isomorphic 8

The Tent Map

1 0.9 0.8

X after one iteration

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1

Figure 3: The Tent Map.

to the typographical system. This is very convenient, because we can now make use of several well known results of Chaos Theory to infer about the FAS. We know that the Tent-map is chaotic i.e. sensitive to initial conditions. What this means is that two very similar FAS with very close initial axioms would contain very different set of theorems. The number of theorems in these two FAS could be very different.

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The Dynamics of FAS

Having established the isomorphism of a typographical FAS with a dynamical system, we are now faced with several questions about the use of such an isomorphism. We wish to know under what conditions does a FAS lend itself to such an isomorphism. We also intend to know whether such an isomorphism would help us in establishing whether an FAS is complete or incomplete. What about the consistency of an FAS ?. Can we test it using this method ?. Finally, we also wish to know whether we can use any quantitative metric for determining the power of a FAS, especially in the context of comparing different FAS which are very similar to each other.

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The study of the above questions constitutes the study of the Dynamics of FAS. By dynamics, I mean the behavior of a FAS under different conditions, such has different choice of axioms (even slight change in axioms may lead to very different theorems) and different choice of rules of inference. We also wish to characterize the power of the FAS. One metric that might be able to capture the power of a FAS is the topological entropy metric (refer to [6] for it’s definition). Topological entropy of a dynamical system measures the degree of freedom the system has in covering its phase space. This, when translated to a FAS would mean the number of theorems it would derive. The number of derived theorems (derived means having a proof) of a certain FAS with certain axioms is one indication of the power of that FAS. However, this is not the only metric. There may be others borrowed from Chaos Theory. May be, the Topological Entropy can be called as the Typographical Entropy for a typographical-FAS.

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Conclusions

To conclude this short paper, I would like to say that much of this paper is speculative and more detailed research need to be carried on the topic. I wish to be able to carry out more detailed analysis of more complex FAS with some of the ideas mentioned in this paper. This is definitely a small baby-step towards uncovering the Dynamics of FAS. The isomorphism of a typographical-FAS with the Tent Map demonstrated in this paper is definitely worth having a closer look. We have not said anything about G¨odel’s Incompleteness Theorem and whether Chaos Theory can show more insights in to this fascinating branch of Meta-Mathematics.

References [1] Kathleen T. Alligood, Tim D. Sauer and James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, 1996. [2] Kurt G¨odel, On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, Dover Publications, Reprint edition, Feb 1992.

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[3] Douglas R. Hofstadter G¨ odel, Escher, Bach: An Eternal Golden Braid, Basic Books, 20th Anniv edition, Jan 1999. [4] James Gleick, Chaos: Making a New Science, Penguin (Non-Classics), Dec 1988. [5] Khalid Sayood, Introduction to Data Compression, Morgan Kaufmann, 1996. [6] http://www.drchaos.net/drchaos/Book/node54.html

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