What is left of physics if the mathematics is removed?

Question

My question is in the title.

It seems to me that (theoretical) physics studies mathematical models of the physical world, and constantly revises them. But isn't studying mathematical models essentially mathematics, too?

I believe that an ontology must be left (e.g. particle physicists have an ontological commitment to elementary particles like electrons).

What else is left?

The question is inspired by the fact that if you open a textbook on advanced physics like quantum field theory, 80-90% of any page seems to be math.

Answer 1

To me it seems plausible to remove mathematics and still have a physical ontology. Science would be less concise, slower, and stunted for starters.

It would begin by replacing every mathematical abstraction with a physical instantiation, just like measurement constants are tied to physical objects or processes. We would recast our theories into these physical representations, and science would lose all it's concision, but would still be comprehensible in theory. Saturn's orbit is not about mathematical equations of spacetime geometry or Newtonian gravity, instead the disk shaped light appearing in our telescopes is the same distance from the Sun as length of a string wound around the Earth in some obtusely long pattern.

This reformulation goes relatively smoothly till we get to reference frames say. A physical instantiation of special, general, or even Galilean Relativity seems quite hard at first. But trivially, I'd offer Galileo's Ship provides a template for mathless conceptions of such problems. What would someone see under such and such physical conditions is all we are after. Given that a person can physically stand in Galileo's ship and drop marbles and watch a pot of water in port and sailing smoothly at sea, or even imagining and/or doing the twin experiment of general relativity, we haven't seemed to encounter the necessity of math yet.

Then we get to quantum mechanics. What is spin physically? How do we understand probability without numbers? This is challenging, but again we are saved by interpretations like Bohmian Mechanics, and possibly Many Worlds and GRW. Plus we can reduce concepts like spin and "spooky-action-at-a-distance" to physical measurement outcomes and not ponder what is going on pre-measurement. Such as, if you vary some tripartite detectors along such and such angles; angles which are physically instantiated say as how long it took to physically spin it at some speed along some axis ("long" and "speed" of which are further physically tied to some other physical standard), there will be physically observible effects in the measurement outcomes. But we know no matter the interpretation, all we are provided are statistical correlations as far as measurement outcomes go. How do we understand statistics, probability, or chance without numbers? I think we can adjust everything to a system of relative credences or bets you would take. Without invoking any reference to numbers you can be sure to take the bet India will win the World Cup Semi-finals over the bet India will win the World Cup. You would always assert a higher relative credence to the former. Does this work for quantum mechanics? I think for Bohmian Mechanics we can say for the double-slit, the physical wavefunction physically bunches up the particles due to things like the no crossing rule and other physical dynamics. So just like the World Cup example, the physical structure of both, semi-finals is before finals, the wavefunction guiding particles toward concentrated and less concentrated areas, gives at least highly tuned relative credences, which can mimic probabilities.

I know there is more to say about QM and this process in general, but maybe this already will be shot down so I don't have to :)

Answer 2

You might still be able to describe a lot of things (not everything), but you will have a hard time to predict things without equations that model the physics. Predictions are what the mathematics is ultimately for, and allows us to know

1. before building a building whether that building can bear its weight
2. before sending a rocket, how much fuel we need and where we need to aim to reach the goal
3. ...

This gets difficult without the mathematics.

Answer 3

It is easy to say that physics would dissappear or be transformed, but those answers are vague and just speculative. So, here's an example how it can be approached. To start, concepts must be precise. So:

Mathematics has two essential functions: it is a language (allows communicating ideas), and it is a tool (allows performing calculations and getting new conclusions).

As a language, it is part of formal languages: languages that are defined in terms of concepts and axioms. Concepts are essentially ideas, and axioms are rules that are applied to concepts and preexisting objective ideas (e.g. Mathematics don't need to define what an object is). Check Kurt Goedel's theorem, is a good example on how concepts and axioms are used as the basis of mathematical formalisms.

As a tool, mathematics provides methods to perform mathematical calculus. Thus, it produces new rules (not anymore axioms, when they are consequences) and even new concepts (e.g. imaginary numbers). Again, Goedel's approach proposes a method of numbering all resulting consequential rules, essentially the result of mathematical calculus.

Physics is a scientific (axioms are empirical truths obtained following the scientific method) discipline (specified by a formal language) of knowledge which studies matter.

Now, the analysis is straightforward.

A) The linguistic dimension of physics is mathematics. So, if mathematics would be removed, physics would just lack of a mathematical representation.

What would left is still knowledge of the physical world, although expressed in other language: either a spoken language or another (e.g. Fortran? tally marks?). So, the second law of thermodynamics would be known, but could simply be expressed narratively (e.g. a cold body would never make a hot object hotter, if they enter in contact). This example uses an informal language (informal for physical purposes), but alternative formal languages exist. See Wikipedia for History of Mathematical Notation.

Consider that physical facts do contribute to mathematics, as the bra-ket notation, which raised from a need of representation of physical facts.

B) The tooling-related dimension of physics, physical calculus, is mostly performed by mathematics. But this as well, can be performed by alternative means. Mathematics turns to be just the most accessible option. But a lot of physical calculations can be performed mentally, without mathematics or by means of physical methods. Farmers don't use mathematics, but tend to optimize the exploitation of physical resources.

Finally, It should be noticed that you assume that math is part of physics, but such idea is just conventional (it depends on a strong boundary between what is physical from what is ideal). In fact, strictly, mathematics and physics study ideal objects, and both are inspired from empirical facts. Physics is said to study matter and energy, but both physics and mathematics are just abstractions of experience. The difference between ideal and physical is considered deprecated in philosophy, modern philosophy tends to accept that all entities are ideal, subjective. In such case, math would not be part of physics, but the opposite: physics would be part of mathematics, the part that is most related with sensibility, while mathematics would be the portion most related to rationality.