**1. Introduction**

What if the universe, on the most fundamental layer, just consisted of numbers? This is a suspicion at least as old as the Pythagoreans. According to Huffman's entry in *The Stanford Encyclopaedia of Philosophy* [1], ". . . in the *Metaphysics*, he [Aristotle] treats most Pythagoreans as adopting a mainstream system in contrast to another group of Pythagoreans whose system is based on the table of opposites . . . . The central thesis of the mainstream system is stated in two basic ways: the Pythagoreans say that things are numbers or that they are made out of numbers. In his most extended account of the system in *Metaphysics* 1.5, Aristotle says that the Pythagoreans were led to this view by noticing more similarities between things and numbers than between things and the elements, such as fire and water, adopted by earlier thinkers." Moreover, according to another contemporary review ([2], p. 14), "according to Aristotle, the Pythagoreans do not place the objects of mathematics between the ideas and material things as Plato does, they say 'that things themselves are numbers' and that 'number is the matter of things as well as the form of their modifications and permanent states'. As the principles of mathematics, numbers are the 'principles of all existing things'."

The importance of number, and more generally, of mathematics, for not only describing but "being" the bricks of the universe was stressed by eminent physicists, like Schrödinger ([3], Chapter III). The introduction of computing machinery creating virtual realities brought these issues to the forefront [4–9]. In a recent bold leap, Tegmark's *Mathematical Universe Hypothesis* [10,11] states that "the physical universe is not merely described by mathematics, but is a mathematical structure". As a consequence, mathematical existence equals physical existence, and all structures that exist mathematically (even in a non-constructive way) exist physically as well.

How could things be numbers? A world "spanned" by numbers can be represented by a single infinite (binary) sequence1, or, equivalently, a single real number.

<sup>1</sup> *A sequence is infinite while a string is finite. A finite prefix of a sequence is then a string.*

In what follows a *number world* will be modelled by a (binary) sequence.<sup>2</sup> Our choice is not to operate with the more geometric-centred Ancient Greek concept of number, which is essential for many continuous models of mathematical physics, but with an algorithmic one which is capable of giving a global perspective of the universe. Adopting this framework is motivated by Plato's mathematical discussion, in *Timaeus*, of the relations between numbers and things, see [2], p. 14 and also [12], and it is adopted here as a matter of *hypothesis*.

All entities encoded therein, including observers as well as measured objects, must be embedded in [6,13]; that is, they must themselves be (formed out of) numbers or symbols [14]. Non-numeric properties associated with such a "world on a sequence" can arise by way of a structural, levelled hierarchy [15].

Epistemologically this can be perceived as "emergence<sup>3</sup> of reality", which is the inverse of reductionism to some more fundamental, basic levels, involving explanations in terms of ever "smaller" entities: physical/universal/natural laws—in particular, relational and probabilistic ones—arise as effective patterns and structures "bottom-up" (rather than "top-down").

Such concepts were quite popular in the *fin de siécle* Viennese physical circles, so much so that they have been referred to as the *Austrian Revolt in Classical Mechanics* [18] and *Vienna Indeterminism* [19]: stimulated by the apparent indeterminacy manifesting in Rutherford's asymptotic decay law and its corroboration by Schweidler [20], Exner's 1908 inaugural lecture as *Rector Magnificus* included the suggestion that ([21], p. 18) "we have to perceive all so-called exact laws as probabilistic which are not valid with absolute certainty; but the more individual processes are involved the higher their certainty". Also Schrödinger's inaugural lecture in Zürich entitled *"What is a natural law?"* adopted and promoted Exner's ideas [22,23], well in accord with Born's later inclinations [24]. Since then classical statistical physics, as well as radioactive decay processes and quantised systems have operated under the presumption that the most fundamental layers of microphysical description are—both theoretically as well as phenomenologically and empirically—consistent with irreducible indeterminism.

Later related ideas have been brought forward in the context of a layered structure of physical theories [15], emergen<sup>t</sup> cognition—perceived as an "emerging epiphenomenon" of neural activity; not unlike traffic jams they arise from the movements of individual taxis [25] –as well as emergen<sup>t</sup> computation [26].

In what follows we shall, in a "Humean spirit" [27], study "laws" as patterns/correlations in sequences using the concept of *spurious correlations in data*, to be defined later. Two guiding theories will be applied: one is algorithmic information theory, the other is Ramsey theory. The gist of these two ways of looking at data is twofold: "all very long, even irregular" data sequences contain "very large" (indeed, as long as you prefer) regular, computable and thus, in physical terms, *deterministic*, subsequences. Secondly, it is impossible and inevitable for any arbitrary data set *not* to contain a variety of spurious correlations; that is, relational properties which could physically be wrongly interpreted as laws "governing" that universe of data.
