Conclusion

There remain a myriad of questions to be asked, each one more provocative than the last. For example: I spoke of the evolution of a theory in order to determine the connotations of the word in question.

However, the question may be asked as to when a theory becomes distinct from the antecedent from which it evolved. Certainly all of us began at the same point from the Big Bang, and so we may all be said to follow one big theory in a certain sense. But that is not how we look at it. At some point, as our theories evolved, they split off from their parents and became mature adults in their own right. There is a certain sense in which we may call relativity a highly evolved version of classical mechanics. Where do we draw the line, then? When is a theory the same as that from which it evolved, and when is it a separate one in its own right? Or is this distinction as illusory as a distinction between ‘good’ fundamental sets and ‘bad’ fundamental sets? Perhaps we need to speak of a continuum of theories, thus making the number of theories in question infinite. However, we do not need this particular continuum, for we already have a generally accepted continuum handy which will work for this purpose: That of time. We shall then speak of a theory at a given time t.

Furthermore, the distinction I made between updating a theory externally and updating it internally is also no clearcut matter; the line is just as blurred as the line between the self and the world—a line which many philosophical perspectives dismiss as illusory. We may save ourselves from the wrath of those holding such viewpoints by considering the distinction to be purely operational and having no deeper connotations.

Then there is the question of how to practically construct fundamental sets. As I have said before, nothing of the sort has ever been done. There have been efforts to axiomatize mathematics (later annihilated by Kurt Godel, of course), but that is not precisely what I am suggesting. Russell attempted to bring together a set of statements from which he hoped he could derive mathematics in its entirety.

In my scheme, we arrive at the scene only after the entire theory has been constructed; after that, we look down onto what we are standing atop and then try to see how far we can reduce it. A small-scale example of such a process is, as has been mentioned before, this very paper. A notion of fundamentality has been constructed in my mind by societal communication. By standing atop this notion, I have attempted to break it down to a sufficiently precise extent.

To some of the more observant ones, I may seem to have done nothing but performed one gigantic cheat in this paper!—for I seem to have done nothing but made the burden of meaning fall on the word ‘indispensable’ instead of ‘fundamental’, lavishly replacing the latter by the former. However, it requires little vision to see that, due to the nature of language, this is the only way the meaning of any word can be conveyed: In terms of other words. I made the meaning of ‘fundamental’ clearer by using a word which has connotations that are not quite as blurry as those of ‘fundamental’, and it has sufficed for our purposes; using it, we have succeeded in reconciling the various seemingly contradictory notions of fundamentality under one satisfactory criterion. The proposal is only bolstered by the fact that, even intuitively, fundamentality and indispensability feel like brothers.

To conclude:

With respect to a given theory at a time T, its fundamental entities are the elements of a set which is both necessary and sufficient for the construction and explication of the theory in its entirety and does not contain any non-necessary elements.

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Source: Aguirre A., Foster B., Merali Z. (Eds.). What is Fundamental? Springer,2019. — 189 p.. 2019

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