Complexity

Volume 2019, Article ID 6427493, 25 pages

https://doi.org/10.1155/2019/6427493

## Mathematics as Information Compression via the Matching and Unification of Patterns

CognitionResearch.org, Menai Bridge, UK

Correspondence should be addressed to J. Gerard Wolff; gro.hcraesernoitingoc@wgj

Received 26 April 2019; Revised 21 August 2019; Accepted 17 September 2019; Published 4 December 2019

Academic Editor: Dimitri Volchenkov

Copyright © 2019 J. Gerard Wolff. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about “information compression (IC) via the matching and unification of patterns” (ICMUP). That is itself a novel approach to IC, couched in terms of nonmathematical primitives, as is necessary in any investigation of the foundations of mathematics. This new perspective on the foundations of mathematics reflects the facts that mathematics is almost exclusively the product of human brains, and has been developed, as an aid to human thinking, mathematics is likely to be consonant with much evidence for the importance of IC in human learning, perception, and cognition. This perspective on the foundations of mathematics has grown out of a long-term programme of research developing the *SP Theory of Intelligence* and its realization in the *SP Computer Model*, a system in which a generalised version of ICMUP—the powerful concept of *SP-multiple-alignment*—plays a central role. This paper shows with an example how mathematics, without any special provision, may achieve compression of information. Then, it describes examples showing how variants of ICMUP may be seen in widely used structures and operations in mathematics. Examples are also given to show how several aspects of the mathematics-related disciplines of logic and computing may be understood as ICMUP. Also discussed is the intimate relation between IC and concepts of probability, with arguments that there are advantages in approaching AI, cognitive science, and concepts of probability via ICMUP. Also discussed is how the close relation between IC and concepts of probability relates to the established view that some parts of mathematics are intrinsically probabilistic, and how that latter view may be reconciled with the all-or-nothing, “exact,” forms of calculation or inference that are familiar in mathematics and logic. There are many potential benefits and applications of the mathematics-as-IC perspective.

#### 1. Introduction

The fundamental nature of mathematics has been a considerable puzzle to mathematicians and others for many years. For example, Roger Penrose writes:

“It is remarkable that

allthe SUPERB theories of Nature have proved to be extraordinarily fertile as sources of mathematical ideas. There isa deep and beautiful mysteryin this fact: that these superbly accurate theories are also extraordinarily fruitful simply asmathematics” ([1, pp. 225-226], bold face added).

In a similar vein, John Barrow writes:

“

For some mysterious reasonmathematics has proved itself a reliable guide to the world in which we live and of which we are a part. Mathematics works: as a result we have been tempted to equate understanding of the world with its mathematical encapsulization. …Why is the world found to be so unerringly mathematical?” ([2, Preface, p. vii], bold face added).

It is clear that, in this quote, the expression “the world” is intended to mean “everything in the observable universe,” in accordance with normal usage. That expression is intended to have the same meaning elsewhere in this paper.

Eugene Wigner [3] writes about “The unreasonable effectiveness of mathematics in the natural sciences”:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which

we neither understandnor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also toour bafflement, to wide branches of learning.” (ibid, p. 14, bold face added).

In this connection:

“… against Wigner’s ‘unreasonable effectiveness’ statement (based on success in the physical sciences) one must ask why maths is often so unreasonably ineffective in the human and social sciences of behaviour, psychology, economics, and the study of life and consciousness. These complex sciences are dominated by non-linear behaviour and only started to be explored effectively by many people (rather than only huge well-funded research groups) with the advent of small personal computers (since the late 1980s) and the availability of fast supercomputers. Some complex sciences contain unpredictabilities in principle (not just in practice): predicting the economy changes the economy whereas predicting the weather doesn’t change the weather” (John Barrow, personal communication, 2017-04-06, with permission).

In keeping with those remarks, Øystein Linnebo writes that “Mathematics poses a daunting philosophical challenge, which has been with us ever since the beginning of Western philosophy” [4, p. 4]. He goes on to say: (1) that mathematics is *a priori* because it seems to be practiced by means of reflection and proof alone, without any reliance on sense experience or experimentation; (2) that mathematics seems to deliver knowledge of truths that are *necessary* in the sense that things could not have been otherwise; and (3) that mathematical knowledge is *abstract*, being concerned with objects such as numbers, sets, and functions, that are not located in space or time, and that do not participate in causal relationships. “In short, by being so different from the ordinary empirical sciences, mathematics is philosophically puzzling; but simultaneously, it is rock solid” [4, pp. 4-5].

This paper attempts to provide some answers. It describes how much of mathematics, perhaps all of it, may be seen as structures and processes for compressing information via a search for patterns that match each other and by the merging or unifying of patterns that are the same (this paper draws on and considerably expands and refines some of the thinking in [5, Chapter 10]). This perspective appears to be novel, not apparently described anywhere in writings about the fundamentals of mathematics (Section 2.2).

The ideas and arguments presented in this paper have grown out a long-term programme of research developing the *SP System*, meaning the *SP Theory of Intelligence* and its realization in the *SP Computer Model*, both of them outlined in Section 3, and both of them founded on evidence that information compression (IC) is a unifying principle in much of human learning, perception, and cognition (HLPC), where “cognition” means such things as reasoning, planning, problem-solving, and the use of natural language.

What appear to be the most compelling kinds of evidence for the importance of IC in HLPC are described in [6, Sections 4 to 21]. Examples include the following: the mismatch between the relatively large volumes of information reaching the retina of the eye and the relatively small capacity of the optic nerve to transmit that information, with evidence for compression of information in the eye; the way in which we merge successive views of a scene to make one; how recognition may be seen as a merging of sensory information with already-stored information; how people with two functioning eyes merge the two simultaneous views of a scene from the two eyes into a single view; how natural language provides an abundance of examples of the “chunking-with-codes” technique for compression of information; and more.

In view of that evidence, and since mathematics has been developed almost exclusively by human brains and as an aid to human thinking, it should not be surprising that mathematics may be founded on compression of information.

The main sections which follow are a summary of some writings about the foundations of mathematics, with a description of the novelty of the idea that mathematics may be understood in terms of IC (Section 2); an outline description of the SP System and its foundations (Section 3); a summary of some related research (Section 4); a description of seven techniques for compression of information which are central in the arguments in the sections (Section 5); the main subject of this paper: how mathematics may be interpreted in terms of IC (Section 6); how similar principles may be seen in the mathematics-related disciplines of logic and computing (Section 7); some remarks about the intimate relation between IC and concepts of probability (Section 8); and Section 9 outlines some potential benefits and applications of the ideas which have been described; Appendix A describes two apparent contradictions of the idea that IC is fundamental in mathematics and related disciplines, and how those apparent contradictions may be resolved Appendix B contains a brief discussion of why we should assume that the future will be like the past.

#### 2. Writings about the Foundations of Mathematics

This section first describes the more prominent “isms” in the philosophy of mathematics and then describes the novelty of the idea that mathematics may be understood in terms of IC.

##### 2.1. Isms in the Philosophy of Mathematics

The variety of “isms” in the philosophy of mathematics testifies to the difficulty of arriving at a satisfactory account of the fundamental nature of mathematics. The more prominent of those isms are summarised alphabetically here:(i)*Formalism.* Linnebo writes: “*Formalism* is the view that mathematics has no need for semantic notions, or at least none that cannot be reduced to syntactic ones” [4, p. 39]. He goes on to describe two versions of formalism and another variant called *deductivism*:(a)*Game Formalism.* “One version of formalism latches on to the comparison of a formal proof with a game played with syntactic expressions. According to *game formalism*, this is all there is to mathematics. That is, mathematics revolves around formal systems, which are syntactical games played with meaningless expressions” ([4, p. 39], emphasis in the original).(b)*Term Formalism*. “As we [define] it, formalism seeks either to banish all semantic notions from mathematics or else to reduce any such notions to purely syntactic ones. While game formalism pursues the former alternative, *term formalism* pursues the latter. Mathematical singular terms are now allowed to denote themselves” ([4, p. 44], emphasis in the original). The gist of what Linnebo says to explain the idea is that something like “6” or “22” is not simply a pattern on a piece of paper, it is a pattern with an associated meaning.(c)*Deductivism*. “*Deductivism* (sometimes also known as *if-then-ism*) is the view that pure mathematics is the investigation of deductive consequences of arbitrarily chosen sets of axioms in some formal and uninterpreted language” ([4, p. 48], emphasis in the original).(ii)*Hilbert’s Ideas.* Linnebo describes David Hilbert’s ideas about the nature of mathematics like this:

“The most sophisticated development of formalist ideas is that of Hilbert’s program. Hilbert proposes … a brilliant strategy of divide and conquer. The way forward, he thinks, is to divide mathematics into two parts.

Finitary mathematicsis a contentful theory of finite and quasi-concrete syntactic types. Hilbert is particularly fond of numerals that take the form of strings of strokes; for example, “|||” is the third numeral. Such numerals are sequences of what we may call Hilbert strokes. Hilbert thinks that finitary mathematics and its foundational axioms can be accounted for using ideas from Kant and term formalism.Infinitary mathematics, on the other hand, is strong enough to describe all of the infinite structures that modern mathematics studies. This part of mathematics can be regarded as a purely formal theory, Hilbert thinks, and when so regarded, can be accounted for by drawing on ideas from game formalism” ([4, p. 56], emphasis in the original).

Linnebo goes on to discuss problems for Hilbert’s program associated with Cantor’s ideas about infinities in mathematics and Gödel’s incompleteness theorems.(iii)*Holism.* About holism, Michael Resnik writes: “The observational evidence for a scientific theory bears upon the theoretical apparatus as a whole rather than upon individual component hypotheses” [7, Location 550], and “… we can construct a so-called indispensability argument for mathematical realism along these lines: mathematics is an indispensable component of natural science; so, by holism, whatever evidence we have for science is just as much evidence for the mathematical objects and mathematical principles it presupposes as it is for the rest of its theoretical apparatus; whence, by naturalism, this mathematics is true, and the existence of mathematical objects is as well-grounded as that of the other entities posited by science” (*ibid.*).(iv)*Intuitionism.* Leon Horsten writes: “Intuitionism originates in the work of the mathematician L. E. J. Brouwer [8], and it is inspired by Kantian views of what objects are [9, Chapter 1]. According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction … Mathematical constructions are produced by the ideal mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician” [10, Section 2.2].(v)*Logicism.* Linnebo writes: “Frege’s philosophy of mathematics combines two tenets. On the one hand, he was a platonist, who believed that abstract mathematical objects exist independently of us. On the other hand, he was a logicist, who took arithmetic to be reducible to logic” [4, p. 21]. And Horsten writes:

“The idea that mathematics is logic in disguise goes back to Leibniz. But an earnest attempt to carry out the logicist program in detail could be made only when in the nineteenth century the basic principles of central mathematical theories were articulated (by Dedekind and Peano) and the principles of logic were uncovered (by Frege). … In a famous letter to Frege, Russell showed that Frege’s Basic Law V entails a contradiction [11]. This argument has come to be known as Russell’s paradox ….” [10, Section 2.1].

An account of Russell’s paradox is in [12].(vi)*Methodological Naturalism.* Alexander Paseau writes: “In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. … Methodological naturalism has three principal and related senses in the philosophy of mathematics. The first is that the only authoritative standards in the philosophy of mathematics are those of natural science (physics, biology, etc.). The second is that the only authoritative standards in the philosophy of mathematics are those of mathematics itself. The third, an amalgam of the first two, is that the authoritative standards are those of natural science and mathematics. We refer to these three naturalisms as scientific, mathematical, and mathematical-cum-scientific. Note that throughout this entry “science” and cognate terms encompass only the natural sciences” [13, Section 1].(vii)*Nominalism.* Linnebo writes: “In contemporary philosophy of mathematics, “nominalism” typically refers to the view that there are no abstract objects” [4, p. 101] and “… we need to do to every scientific theory what we did to finite number ascriptions, namely to “nominalize” the theory by reformulating it in a way that avoids all commitment to abstract objects” [4, p. 105].(viii)*Platonism.* In the “Platonism” view, mathematical entities “are not merely formal or quantitative structures imposed by the human mind on natural phenomena, nor are they only mechanically present in phenomena as a brute fact of their concrete being. Rather, they are numinous and transcendent entities, existing independently of both the phenomena they order and the human mind that perceives them” [14, pp. 95-96]. Such ideas are “invisible, apprehensible by intelligence only, and yet can be discovered to be the formative causes and regulators of all empirical visible objects and processes” ([14, pp. 95]).(ix)*Predicativism.* Horsten writes: “The origin of predicativism lies in the work of Russell. On a cue of Poincaré, he arrived at the following diagnosis of the Russell paradox. The argument of the Russell paradox defines the collection **C** of all mathematical entities that satisfy . The argument then proceeds by asking whether **C** itself meets this condition, and derives a contradiction. …” [10, Section 2.4].(x)*Realism.* Resnik writes: “My realism consists in three theses: (1) that mathematical objects exist independently of us and our constructions, (2) that much of contemporary mathematics is true, and (3) that mathematical truths obtain independently of our beliefs, theories, and proofs. I have used the qualifier “much” in (2), because I do not think mathematical realists need be committed to every assertion of contemporary mathematics” [7, Location 84].(xi)*Structuralism.* Linnebo writes: “Structuralism is a philosophical view that emphasizes mathematics’ concern with abstract structures, as opposed to particular systems of objects and relations that realize these structures. Consider three children linearly ordered by age and three rocks linearly ordered by mass. These two systems of objects and relations realize the same abstract structure, namely that of three objects in a linear order. All that matters for mathematical purposes, according to structuralism, is the abstract structure of some system of objects and relations, not the particular natures of these objects and relations” [4, p. 154]. There is more about structuralism in Section 4.4.

##### 2.2. The Novelty of the Idea That Mathematics May Be Understood in terms of IC

With regard to the idea that mathematics may be understood in terms of IC, three recent books about the philosophy of mathematics [4, 15, 16], an article about the “Philosophy of Mathematics” in the *Stanford Encyclopedia of Philosophy* [10], two near-recent books in the same area [7, 17], and one recent book on mathematics-related areas [18], make no mention of anything resembling IC. More generally, the idea that IC might be part of the foundations of mathematics appears to have no place in any of the isms in the philosophy of mathematics (Section 2.1), or any other writings about the nature of mathematics.

Devlin’s academic book, *Logic and Information* [19], aims to develop a mathematical theory of information, a goal which is related to but distinct from the central idea in this paper, that mathematics may be seen to be largely about IC.

A book for nonspecialists by Devlin, called *Mathematics: The Science of Patterns* [20], discusses things like “patterns of symmetry [such as] the symmetry of a snowflake or a flower” (p. 145) (where “symmetry” implies redundancy, which is an important part of IC) and “the patterns involved in packing objects in an efficient manner” (p. 152) (where “efficient” may be seen to relate to IC). But, these kinds of patterns are quite different from the concept of an “SP-pattern” in the SP System (Section 3.2), and IC in the foundations of mathematics is not made explicit or discussed.

Resnik’s academic book on *Mathematics as a Science of Patterns* [7] is discussed in Section 4.4.

Amongst isms in the philosophy of mathematics (Section 2.1), the one which is perhaps most closely related to the thesis of this paper is *intuitionism*, meaning that mathematics is a creation of the human mind. Clearly, the invention and development of mathematical concepts has been done mainly by human brains, and they are designed to assist human thinking. But, there appears to be no recognition in intuitionism of IC as a unifying principle in HLPC or mathematics, of unsupervised learning, or of the representation or knowledge with structures like SP-multiple-alignment.

Concepts in the SP System also relate to structuralism, as discussed in Section 4.4.4.

#### 3. Outline of the SP Theory of Intelligence and the SP Computer Model

As noted in the Introduction, much of the thinking in this paper derives from the SP System, meaning the *SP Theory of Intelligence* and its realization in the *SP Computer Model*. This section describes the SP System in outline, with sufficient detail to allow the rest of the paper to be understood.

In most papers in the SP programme of research, including this one, it has proved necessary to provide a section like this one, or an appendix, which provides an outline of the SP System. This is to ensure that each paper is free standing and can be read without the need to look elsewhere for information about the SP System.

The most comprehensive account of the SP System is in the book *Unifying Computing and Cognition* [5], which includes a detailed description of the SP Computer Model with many examples of what the model can do. A shorter but fairly full description of the SP System and its strengths and potential is in [21]. Details of these and other publications, including several papers about potential applications of the SP System, may be found, with download links, on http://bit.ly/2Gxici2.

Source code and Windows executable code for the SP Computer Model may be downloaded via a link under the heading “SOURCE CODE” on the same page.

##### 3.1. Foundations

The overarching goal in developing the SP System is, in accordance with Ockham’s razor, the simplification and integration of observations and concepts across artificial intelligence, mainstream computing, mathematics, and HLPC, with IC as a unifying theme.

Since people often ask what the name “SP” stands for, it is short for *Simplicity* and *Power*. This is (1) because “simplification and integration of observations and concepts” means the same as promoting simplicity in one’s theory whilst retaining as much as possible of its descriptive and explanatory power and (2) because within the SP Theory, compression of a body of information, **I**, means maximising the simplicity of **I** by reducing, as much as possible, repetition of information or *redundancy* in **I**, whilst retaining as much as possible of its nonredundant descriptive or explanatory power.

Despite the ambition of attempting simplification and integration across AI, computing, mathematics, and HLPC, much has been achieved: the SP System combines simplicity—in being largely composed of the relatively simple mechanisms for creating and processing SP-multiple-alignments (Section 3.3) and it exhibits descriptive and explanatory power in modelling diverse aspects of intelligence, in accommodating diverse kinds of knowledge, and in their seamless integration in any combination, as outlined in Section 3.7.

The idea that IC might be significant in the workings of brains and nervous systems was pioneered by Fred Attneave [22], Horace Barlow [23, 24], and others, and it has been developed in many other studies, many of which are outlined and referenced in [6, Section 3].

##### 3.2. The Main Features of the SP System

As shown schematically in Figure 1, the SP System is conceived as a brain-like system that receives *New* information via its senses and stores some or all of it, in compressed form, as *Old* information.