Extending abstract linear propositions not bound by the strict principles of causality to formulate physical theories that are non-linear and causal, is a problem of mathematical physics. Generalizing the part to cover the whole is against natural principles. It works in scalar numbers, which follow a general principle for accumulation and reduction. But it won't work in physics, where the conditions are not linearly additive or reductive.
Let us take one example given by a friend of the collection of electron orbitals for the carbon atom is the domain for a singular function Q. Wherever f(x) in set Q is = f(y) in set P, a hydrogen atom, then there is a natural implication that x=y, meaning that an electron in the carbon atom with corresponding or analogous values to an electron orbital in the hydrogen atom as imputed by internal or external condition that exceeds the parameters of the sets Q and P, x=y can become an independent position. In this case of electrons, the spectrum of binding could increase to function with more complex synthesis to give rise to organic life.
Hydrogen has only one shell K and one electron in a spherical orbital 1s close to the nucleus. It consists of only a single proton - atomic number = mass number = 1, and its mass is 1.007825 amu. It is like a book - an element of a set.
Carbon has two 6 electrons arranged 2 each in 1s, 2s and 2p orbitals. Shell number one can only hold 2 electrons, shell two holds the rest 4. Atomic number of carbon is 6 and mass number of carbon is 12.1. It is like a library - a set of books.
But a book is not a library. A set with a single element is void. Otherwise it will lead to Russell's paradox. The shells and orbitals are not just higher numbers. They are not linearly additive functions. They have different physical properties.
Though hydrogen and carbons belong to a set of atoms, when we go into details, they are different classes like apples, mangoes, oranges, lemons, grapes, berries, are all fruits, but are not the same. The non-linearity in the above example comes from the nature of shells. The first shell can hold up to two electrons, the second shell can hold up to eight (2 6) electrons, the third shell can hold up to 18 (2 6 10) and so on. The general formula is that the nth shell can in principle hold up to 2(n^2) electrons. The moment you introduce higher number of cells, we introduce exponential terms that breaks linearity.
Hence, if we say that the collection of electron orbitals for the carbon atom is the domain for a singular function Q. Wherever f(x) in set Q is = f(y) in set P, a hydrogen atom, then there is a natural implication that x=y, meaning that an electron in the carbon atom with corresponding or analogous values to an electron orbital in the hydrogen atom as imputed by internal or external condition that exceeds the parameters of the sets Q and P, x=y can become an independent position - it is not only misleading, but also outright wrong.
In this case of electrons, the spectrum of binding could increase to function with more complex synthesis means they belong to different classes not subject to additive or reductive process.
Parameters of function can be exceeded under the right conditions. But unless those conditions are taken into account, abstract mathematics cannot explain physics.