Tertiary Considerations

Going back to vectors: We may consider updating our theory to include or exclude an object to be analogous with adding or subtracting a vector to our n-dimensional theory-vector.

Let us turn towards the fact that there are two kinds of vectors that may be added: One that has a component orthogonal to all the n dimensions of our theory-vector, and one that is writeable in terms of the n-dimensional basis.

This dichotomy has some important implications.

Adding a vector without any orthogonal component corresponds to updating our theory in a manner such that the update, whatever it may be, was something that was derivable from the fundamental set that was at hand without any external help or knowledge. In other words, it corresponds to updating a theory by means of introspection: An internal update that was already implicitly present.

Adding a vector that does have an orthogonal component is a bigger step. It refers to an update that was not derivable from the fundamental set that was at hand. We needed something external.

You are a mathematician. You have just constructed a proof for Fermat’s Last Theorem. One week after you first thought of it, as you were working out the finer points on your way home from McDonalds’, you realize that there is a flaw. This is an update of the first kind: The flaw was present all along, and you required no extra knowledge or experiments to know of its existence. Just some introspection. Some may even say that, in some sense, you knew that this flaw existed, and that it merely did not come up to conscious reflection until now. This is an update wherein the vector added had nothing orthogonal to the vector corresponding to the previous theory.

Suppose, now, that you are a biologist attempting to ascertain whether a tomato is a vegetable or a fruit. You examine it under a microscope and observe certain telling features enabling you to classify it as a fruit.

I hope I have made this rather subtle distinction clear. The evolution of a theory can be accounted for in terms of these two phenomena.

We also often speak of degrees of fundamentally. To account for this disposi­tion, we need to consider the real world situation in all its ambiguity and apply our reflections to such a situation. To the best of my knowledge, no theory of practical use has yet been constructed such that we could explicitly pick out its fundamen­tal sets. Even when it comes to the relatively straightforward Newtonian theory of mechanics, there is much more to it than just forces and bodies. We still do throw the word fundamental around with reference to them with a great degree of confidence, however.

One obtains some notions of what is more indispensable and important and what is less while in conversation by our implicit observations: If I see that I am able to explicate a greater number of things with the help of a given object, it obtains a greater degree of fundamentality. This is how the word ‘fundamental’ is used in everyday communication.

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