A fundamental question about relations between axioms

Question

In mathematics, there are several sets of axioms. For example, we have ZFC axioms, Peano axioms, Hilbert's axioms of Euclidean geometry(https://en.wikipedia.org/wiki/Hilbert%27s_axioms), and so on.

My understanding of axioms is as follows,

Once a set of axioms has been determined, one system has been established. For example, based on Hilbert's axioms, we develop Euclidean geometry in a rigorous way. However, each set of axioms can be derived from a more basic/fundamental set of axioms, by defining their terminology properly.

For example, in set theory, we define $0 := \emptyset$ and $1:= \{\emptyset\}$, and so on. By doing this, we can define the natural numbers and show this set of natural numbers defined satisfies Peano axioms. Peano axioms become the theorems in terms of set theory.

Also, if we define points as $(x,y) \in \mathbb{R}^2$, straight lines and so on, we can show this system satisfies the axioms of Euclidean geometry. As far as I remember, Hilbert said the other way around, like "Geometry can be reduced to analysis and analysis can be reduced to arithmetics." (I don't remember the exact wording, sorry) That's how the rigorization/axiomatization started in the late 19th century/the early 20th century.

Here, my curiosity extends to physics. In physics, there are also different systems, such as Newtonian mechanics, quantum mechanics, relativity and so on. In each system, there is a set of laws, which Newton called axioms and he even considered geometry as part of mechanics.

Let's take just Newtonian mechanics. There are three laws. Are these laws considered to be axioms in a mathematical sense? (To be honest, it doesn't seem to me that mathematics and physics are separate. I don't think even the great mathematicians/physicists such as Galileo, Newton, Einstein, didn't think that way, too.) So, could the laws in Newtonian mechanics, for example, the famous $F = ma$, be deduced from set theory by defining the proper terminology such as Force, mass, velocity and so on, as we can do it with Peano axioms and Hilbert's axioms?

Lastly, even though they are not deduced by set theory, does set theory or a rigorous set of mathematical axioms have to be with them if they are dealt with rigor? I am not sure how to view the laws or principles in physics in a rigorous mathematical point of view. Any insights into or help with this would be very much appreciated! Thank you very much.

Answer 1

Once a set of axioms has been determined, one system has been established. For example, based on Hilbert's axioms, we develop Euclidean geometry in a rigorous way. However, each set of axioms can be derived from a more basic/fundamental set of axioms, by defining their terminology properly.

So you say that the axioms of any theory $A$ can always be derived from the axioms some "more basic" theory $B$. Of course that means $B$ can derived from some more basic theory $C$, which derives from $D$, which derives from ... I hope you see the problem here.

What you are talking about is actually not deriving $A$ from theory $B$, but rather modeling theory $A$ within theory $B$. Exactly because you cannot keep defining things in terms of previously defined things, but must start somewhere, a mathematical theory starts with a collection of "undefined" primitive terms, and a collection of statements ("axioms") describing how those terms relate to each other. By this, the axioms take the place of definition, in a sense "defining" the primitives in terms of each other. And together, the axioms and primitives define the mathematical theory. They determine what the theory is about, and what is true or false within it. One has to be careful in choosing the axioms, because a poor choice can result in a contradiction, and anything can be proven from a contradiction. So in such a theory, every statement is both true and false. Such a theory is inconsistent. Theories without a contradiction are consistent.

Now, within a different theory $B$, we can sometimes define the primitives of $A$ in terms of the primitives of $B$, and use those definitions to prove that the axioms of $A$ are all true in $B$. This is not deriving $A$ from $B$, but rather modelling $A$ in $B$. The definitions used are our choice, not something that "must be". For example, you can model the Peano axioms in set theory by defining the successor function as either $s(n) = \{n\}$ or as $s(n) = n \cup \{n\}$. The sets acting as natural numbers are radically different, but either approach works, allows you to prove the Peano axioms from the definition.

So modelling doesn't derive the theory. What it does is tell you that if $B$ is consistent, then $A$ has to be as well, because a contradiction in $A$ would then by model also be a contradiction in $B$.

Here, my curiosity extends to physics. In physics, there are also different systems, such as Newtonian mechanics, quantum mechanics, relativity and so on. In each system, there is a set of laws, which Newton called axioms and he even considered geometry as part of mechanics.

It has only been in the last ~150 to 200 years that mathematics and physics have been recognized as two separate fields. Before then, it was not that physical laws were considered "axioms", but rather that axioms were considered to be "laws" - statements about nature that we believe are true, but can only substantiate empirically. The description I gave above about axioms being statements we use to define a theory was not part of their thinking. One could no more claim that there was more than one line through a point parallel to a given line, than one could claim that objects dropped on Earth do not fall.

And of course, that was the problem. First, some mathematicians working to prove the parallel postulate found that you apparently got a consistent theory if you assumed it was false. Then Weierstrass had to go off and show that a model for a part of this theory could be built within Euclidean geometry. And Poicare one-upped him by modelling the entirety of hyperbolic geometry within Euclidean geometry. Suddenly accepting that the parallel postulate was true also meant accepting that it can be false instead. This was a matter of choice, not something that just "is". And if it is the case for one axiom of mathematics, why should it not be the case for the rest? And so Mathematics and Physics got an amicable divorce and went their separate ways.

Let's take just Newtonian mechanics. There are three laws. Are these laws considered to be axioms in a mathematical sense? (To be honest, it doesn't seem to me that mathematics and physics are separate. I don't think even the great mathematicians/physicists such as Galileo, Newton, Einstein, didn't think that way, too.)

Galileo, Newton, and many others lived before this separation. Einstein lived after, and as part of his development of General Relativity studied geometries quite deeply. He was well aware of why Math and Physics broke up.

So, could the laws in Newtonian mechanics, for example, the famous $F = ma$, be deduced from set theory by defining the proper terminology such as Force, mass, velocity and so on, as we can do it with Peano axioms and Hilbert's axioms?

You can take the laws of physics as axioms of a mathematical theory. Effectively, this is exactly how physicists (or any other scientist) applies mathematics to their field. The physics terminologies such as "force, mass, distance, time" are primitives and the laws are the axioms that establish the theory. So "physics" becomes a mathematical theory. But if that is what you do, then ALL you have is a mathematical theory - a bunch of results about abstract concepts without any "real-world" meaning. It doesn't become actual physics until you step out of that theory and decide that "force" is what happens when you press on something, what pushes against the ground, and what the ground applies to hold you up.

In "Surely You Are Joking, Mr. Feynman", Richard Feynman discussed his trouble teaching some students who had the theory of optics down pat. But when he tried to get them to apply the very results just discussed to a real world situation, they would give the wrong answer, because they had apparently learned the theory as if it were only mathematics, not physics, and when thinking of the real world, they never thought to apply it, but went with their misguided intuition instead.

I am a mathematician (at least, at heart). I love mathematics. I love the freedom to build "castles in the air" as I please. Anything I build is, and always will be, mathematics. Physics is different. A theoretical physicist may spend years working in a particular theory, only eventually to have experimental results disagree. All the work done ceases to be physics at that point. It is still mathematics, and still may be useful in other areas, but it is no longer physics.

Lastly, even though they are not deduced by set theory, does set theory or a rigorous set of mathematical axioms have to be with them if they are dealt with rigor? I am not sure how to view the laws or principles in physics in a rigorous mathematical point of view. Any insights into or help with this would be very much appreciated! Thank you very much.

Even if you build a mathematical theory of physics within the set theory or elsewhere, it does absolutely nothing to establish the truth or falsity of the physics. Having no logical contradictions is important for a physical theory, but what establishes a physical theory is its usefulness in describing the real world - in particular, in making predictions about that world that turn out to be true. Engineers do this constantly.

The difference between mathematics and any science is that mathematics establishes its truths deductively - logically proving results from the axioms. The limitation to mathematics is that it does not establish that $1 + 1 = 2$. Instead, it establishes that "under the Peano axioms, $1 + 1 = 2$". On the other hand, science establishes its truths empirically - by making predictions over and over again, and showing that the statements have always held true every time they've been tried and the results carefully established. The limitation to this approach is that you can never be completely sure that the next time will also be true.

You might have an extraordinarily sucessful theory about mechanics that just works everywhere and has everyone lauding your name as the greatest ever. Then some guy plays with some equations and up pops a fixed speed for electromagnetic radiation, contrary to your theory's prediction that all speeds are relative. And while everyone is working to explain why your theory must still be right, some other doofus says "well maybe it is constant", and then someone tests it, and next thing you know, that doofus is declared right and goes on poking more holes in your theory, and everyone is proclaiming him to be the greatest.