What is the relation between pure mathematics, applied mathematics and Cellular Automata?

I was reading about pure mathematics and I am wondering if there is a relation between it and Cellular Automata.

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.

For example, elementary cellular automata transforms an input of {3 cells} based on a rule set to a fractal like rule 90, how this will be described in pure mathematics perspective?

Another fractal like Pascal's Triangle can determine the coefficients that arise in binomial expansions, how it can be related to applied mathematics?

Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry.

The use here of Pascal's triangle is determining the coefficients of binomial expansion, so how it can be related to applied mathematics?

• "Pure" and "applied" mathematics aren't rigidly defined concepts. I would say that a cellular automaton is pure mathematics: it's a mathematical object that can be studied entirely by doing mathematics. – David Richerby Oct 11 '15 at 18:40
• What do you mean when you say that a cellular automaton is based on a rule set to a fractal? Rule 90 is only a fractal for one input, and many ECAs display no fractal behaviour. Binomial expressions have many applications: they tell you the coefficients of $(a+b)^n$ and they tell you how many ways you can choose $k$ elements from a set of $n$ items, concepts with diverse applications. Why do you suggest they cannot be related to applied mathematics? – Lieuwe Vinkhuijzen Oct 11 '15 at 19:03

The definitions given in Wikipedia are trying to convey a very abstract concept by means of examples, which, to my taste, isn't satisfactory.

Thus, before answering this question we'd need to ask ourselves why are there two in the first place. And if there is a real need for two, then:

How is mathematics (both pure and applied) possible? -- Immanuel Kant

History

The answer to the first question appears to be very involved. In European tradition, the question goes back to Plato's distinction between the ideal and the concrete. These roughly translate into analytic and synthetic in modern philosophical language. Plato ultimately believed that mathematics is discovered rather than constructed, but this still puts mathematics in the realm of ideal due to Plato believing that only the ideal actually exists, and that the concrete is but an imperfect reflection of the ideal. This was in stark contrast to the existing tradition of viewing mathematics as a practical skill of measuring and counting.

Since that time on, multiple attempts were made to either unify both, or to define one in terms of the other, or to completely dismiss one of the two. Ultimately, in the beginning of the 20'th century the argument boiled down to the ability of defining the classical mathematics. This split the philosophy of mathematics into logicism (with Frege as its forerunner), formalism (with Hilbert as its advocate) and psychologism (intutionism), the idea advanced by Brouwer. Somewhat under-represented here are the universalist views held by Klein (no good online quotes, sorry). The later is the view that synthetic and analytic truth are the same, and that when a seeming contradiction arises, it is probably because we either don't have a proper observation technique, or that we don't have a proper mathematical explanation for the observation.

The Problem

It all boils down to the question of what is truth, and are there different kinds of truths? I will first give an example of the problem and then discuss it.

Stating that

$$\textit{bachelors are males}$$

Imposes a restriction that the interpretation of "bachelors" must be a subset of the interpretation of "males", but this restriction doesn't result from logical construction of the sentence, rather it is our experience (and the reasoning non-included in this sentence) that all bachelors we've seen so far, and are likely to meet in the future will be male. This is an example of synthetic truth. While in this example it may look like it could be possible to resolve the problem by adding more logical predicates, the problem is in fact unresolved as of now. More generally, this can be seen as the disparity between material implication and "real" implication.

Using material implication, typically denoted as $\implies$ we can obtain:

$$\textit{water is wet} \implies \textit{sky is blue}$$

And while the sentence is true in a pure logical sense. From the perspective of non-logical, or extra-logical reasoning it is false, since the perceived properties of water have no immediate impact on the perceived color of the sky.

Lewis was, perhaps, the first to notice and try to address this general problem by creating more restricting logical systems. Another way to approach this problem, and the stance taken by many contemporary philosophers is laid out in Two Dogmas by Quine. To quote the later:

Physical objects, small and large, are not the only posits. Forces are another example; and indeed we are told nowadays that the boundary between energy and matter is obsolete. Moreover, the abstract entities which are the substance of mathematics -- ultimately classes and classes of classes and so on up -- are another posit in the same spirit. Epistemologically these are myths on the same footing with physical objects and gods, neither better nor worse except for differences in the degree to which they expedite our dealings with sense experiences.

Put simpler: "pure" mathematics is just a convenience, a make-believe thing that doesn't have any factual content and has no bearings on what truth is. Surprising as it may sound, we still use the terminology in its pre-Quinean sense.

Finally

When forced to define abstract and applied mathematics, I'd say that:

1. Applied mathematics is pure mathematics restricted to the synthetic truth. I.e. the kind of mathematics that can be verified by experimentation.
2. Pure mathematics is the art of discovery of patterns. The parallel between art and mathematics is not new, and, in fact, Hardy, the iconic pure mathematician holds that:

I am interested in mathematics only as a creative art.