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So usually one maps a math equation to an ontology in physics. Imagine me modelling a ball rolling up an inclined plane at an arbitrary angle. Now, the moment I make the inclined angle 90 degrees to the plane it collides with the plane and becomes a turning point. This can be modelled as an elastic collision (0 time duration and infinite force). This trick is known as renormalization.

Now, I can have an ontology that says all interactions must have finite durations the moment I choose the inclination angle to be 90 degrees this mapping is void. What happened? Well, we mapped force to a number in this ontology and the moment we let force be infinite we remember infinity is a concept not a number and thus we can arguably extended our ontology.

Question

What all mathematical operations leave the ontology invariant for a physicist?

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  • Also does this suggest the math to be more fundamental than ontology? Commented Nov 1, 2023 at 10:31
  • ???? A math theory assumes the existence of some specific objects and structures and it applies to models that satisfy the axioms of the theory. Commented Nov 1, 2023 at 11:25
  • Yes. All my question is saying that physicists extend the ontology (set of axioms) by using a concept (renormalization) Commented Nov 1, 2023 at 11:31
  • 90 degrees to what? What collides with what plane? What becomes a "turning point"? I cannot make head or tail of this question. Commented Nov 1, 2023 at 13:22
  • What is the "mathematical operation" here? Physicists are constantly pushing the boundaries of their models to accommodate new effects, idealized situations (as in this case), etc. There is no telling out of context and ahead of time which model restrictions can be profitably relaxed or removed. This does not suggest that mathematics is "more fundamental" than ontology, it suggests that it is more flexible than the fragment of it that any particular model uses. Which should not be surprising because it is developed to be this way. The maxim is to maximize options, as Maddy puts it.
    – Conifold
    Commented Nov 1, 2023 at 13:22

1 Answer 1

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I suspect you are confusing ontology with simplifying assumptions. Theoretical physics consists of a large number of mathematical models of reality, each of which has limited applicability and most of which make one or more simplifying assumptions about reality. When you switch between one theoretical model and another, you are not committing yourself to a different belief about what actually exists- you are making a different set of simplifying assumptions which better suit the problem you are trying to address. Take an analogy with cartography. You cannot depict all aspects of the Earth's three-dimensional surface on a flat piece of paper, so to draw a map of the world you need to sacrifice some aspect of reality, and you adopt one out of a number of possible projections depending on your purposes. Drawing one or other form of map is not a matter of adopting a new ontology- you are not changing your mind about what reality is but about how to illustrate it.

In a similar way, different mathematical models are used for different purposes in physics, and not all of them are compatible. If you want an exhaustive list of which aspects of theoretical physics are compatible with which other aspects, then you are asking for too much. However, at a high level you can take it that most theoretical models ideally in use in physics are left unchanged by a range of transformations, such as rotations, translations etc, which reflects an underlying symmetry in reality.

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  • In the use case in the example I give of this "confusing ontology with simplifying assumptions." Commented Nov 1, 2023 at 20:02
  • I can reformulate the axioms such that one of the them say interaction duration should be finite Commented Nov 1, 2023 at 20:07
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    Sure, they are just different simplifications. In reality, the collision between the ball and the vertical ramp is hugely complicated, involving forces between countless trillions of particles. We simplify it by assuming the ball is perfectly elastic, perfectly spherical, can be treated as a point mass, rebounds instantaneously etc etc. We don't believe the ball really does have those properties. Commented Nov 1, 2023 at 20:29
  • I used the word model and can model it's such. QfT is spammed with renormalization. With our classical intuition surely the naive idea of simplifying won't work Commented Nov 2, 2023 at 4:53

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