Indispensability

With this variable in place, we may finally pose a well-defined question: “With respect to a given theory, how may we determine those objects which are fundamental?” And furthermore, what does it mean to call a given object fundamental?

Let us examine an object which a large part of the scientific and philosophic community holds to be highly fundamental in nature: Mathematics.

Mathematical objects blatantly fail spatiotemporal existence. Despite this, its fun- damentality is deeply believed in. There have been numerous arguments attempt­ing to prove its a priori existence. Let us take, for example, Hilary Putnam’s and W. V. O. Quine’s indispensability thesis [3]. The argument ran thus:

1. We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

2. Mathematical entities are indispensable to our best scientific theories.

3. We ought to have ontological commitment to mathematical entities.

This seems to me to point the way towards the answer to our query; our solution­statement suggests itself thus: That which is indispensable to a theory is fundamental to it. I shall show how this, in one fell swoop, integrates all our scattered notions of fundamentality.

An entity may be said to be indispensable to a theory if it is necessary for the complete explication of that theory; if one cannot explicate the theory in terms independent of such an entity. The physical dimensions, mathematical implications, etc. of the entity do not matter, for they have no direct influence on its indispensability to the theory; indispensability is an abstraction birthed from language. If a proponent of the theory claims that he cannot describe his theory without referring to a certain entity, that entity is indispensable to it; it is fundamental to it. (Something to note here is that it is only the proponent himself who is in a position to decide which entities are required and which ones are not.)

It follows that we are not making any ontological comments on the nature of fundamentality but purely epistemological ones, because fundamentality becomes entirely determined by and dependent on communication and language.

“Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt.” The limits of my language mean the limits of my world.

Some reflection exposes this statement to be tautological in nature (something which would greatly appeal to Wittgenstein, for he is one who has maintained that all the statements one can make about the world are tautologies). That which we cannot describe, we cannot comprehend. Language is antecedent to everything. As far as an individual as concerned, nothing that transcends language can be said to exist; it cannot even be said to not exist, for it fails description. It is simply beyond the boundaries of the individual’s logic.

Coming back to Indispensability: It is known that one cannot reduce a theory down to a set of independent statements; that a theory is a set of interconnected, interdependent statements that lose meaning when isolated from one another. To speak of forces is meaningless without speaking of bodies in parallel; to speak of bodies is meaningless without speaking of forces in parallel. And so, in accordance with our reduction of a theory with respect to indispensability, we may reduce a theory to nothing less than a set of fundamental entities: A fundamental set whose interdependent, inter-determining elements would be the fundamental entities. This set would then be a necessary condition to describe the theory in question. (I do not even claim such a fundamental set to be unique to a theory, something which will become evident as we move on.)

But we must also take care not to admit too much in: This set’s elements should not only be necessary and sufficient, but there should also not be a single non-necessary entity, for otherwise, we would not be true to the intuitive notion of fundamentality we have by letting a horde of other non-necessary entities into that class.

The fundamental entities in a theory are, then, that core set of pointers required to describe a theory.

To refrain from Hegelian labyrinths in communication is always a virtue. In this spirit, I shall illustrate my point with one of the most simplistic mathematical frame­works known to us; it is its very simplicity that heightens its illustrative power. Let us represent our theory as an n-dimensional vector. When resolved, the orthonormal vectors we obtain are analogous to the fundamental entities of that theory.

When working with vectors, we have the freedom to select any arbitrary basis. It is known that there are an infinite number of other basis (with increasing convolution which make them harder to work with) that have the same representational power as (x, y). Correspondingly, it is the case that there exist an arbitrarily high number of fundamental sets to choose from from which we may construct our theory. Selecting a basis is analogous to selecting a fundamental set.

It is known that back when quantum mechanics was still young and busy clob­bering physicists over their heads with its shocks, Erwin Schrodinger and Werner Heisenberg developed, in a roughly parallel manner, two completely independent and equally powerful mathematical representations for it: Heisenberg’s matrix­mechanics and Schrodinger’s wave mechanics.

However, as far as conversations go, it is wave mechanics that dominates; when one explains quantum mechanics to a layman, it is wave mechanics that is explained. Why do we instinctively go to this particular explanation, despite the fact that matrix mechanics does not lack in comparison to it in any way?

We do this due to the simple reason that wave mechanics is easier to deal with and communicate as opposed to its matrix counterpart. Why it is easier to communicate using waves is a different question altogether; presently, all I am concerned with is the fact that it is easier to communicate using them.

And so similarly, our selection of the fundamental set is based on its relative ease of communication and computation; and as a result, our choices of the fundamental set generally end up converging.

The analogy with vectors happens to be quite extensive—the dot product of two theory-vectors can tell us how similar two theories are, while the cross product may be said to give a third theory based on the previous two but yet distinct from them, for e.g. quantum biology from quantum mechanics and biology—but exploring it further is not relevant to our current purposes.

There a very interesting observation to be made here, a bootstrapping-like phe­nomena occuring: We obtain our theory vector first and work our way down resolving it to see what it is made up of. The observant reader may have noted that this is, in fact, exactly what is being done in this article! I am standing atop our everyday notion of fundamentality in order to define that very notion more precisely. These bootstrappings happen to occur quite frequently in language, although dissecting the workings of such phenomena will also take us away from our agenda.

Now, when it comes to the word fundamentally, there is an added quirk: We speak of theories themselves being more fundamental than one another! How do we account for this?

The same process and product suffices. What is the main aim of any theory? To explicate a certain set of phenomena, we have said previously. Therefore, a theory may be said to be more important, or to be more indispensable, or to be more funda­mental, with respect to a given question we wish to answer. If we are looking for a framework which will allow us to make physical predictions—if all we are bothered about is the empirical behavior of the Universe—physics satisfies the criterion suf­ficiently, and we may call physics more fundamental than numismatics. Otherwise, depending on the specifications on our quest, it may be logic, or mathematics, or philosophy. And so on.

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