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"The difference between math and physics is that physics describes our universe, while math describes any potential universe"
This was one of my math professor's arguments in trying to convince me to major in math over physics. I thought it was an interesting concept, so I was wondering
Known writings or your own opinions are appreciated, thanks!
I think what your professor meant by saying; "The difference between math and physics is that physics describes our universe, while math describes any potential universe" is that the study of Pure Mathematics provides a deeper and more substantial understanding of reality than physics.
Im guessing that your Professor is in some sense a Platonist, as Philosophy of Mathematics is grounded in Plato's theory of the forms. Mathematics and Physics present two distinctively different ways to understand reality, and the distinction is in the different approaches to discerning what reality is in essence by discerning what it is ultimately reducible to.
The Physicist will tell you its matter all the way down, that reality in essence is matter, and that all matter is ultimately reducible to a particle which is indivisible; the materialist approach. Physics will give you technical expertise, but tell you little about what reality is in essence as materialism is an assumption.
The mathematician will generally describe reality in terms of the forms, that reality is ultimately reducible to principles of logic, the idealist approach. Pure Mathematics will give you a deep understanding of the nature of reality and technical expertise.
I believe this distinction between the different intellectual fruits of labour for both fields was at the core of your professors arguement.
Math doesn't describe a universe any more than a hammer describes a skyscraper. It's just a tool. From the mathematician's point of view, if the physicists find it useful, then good for them ... but math doesn't care one way or the other. On the other hand, the greatest of the mathematicians have been great physicists. Newton and Gauss for two. So there's some kind of symbiosis between math and physics. But math doesn't describe "the" universe and it also doesn't describe "a" universe. If I give you the number 5, that's math. If I tell you 5 miles per hour, that's physics. If I say 5 trees, that's forestry. Math is a tool. On its own terms and in isolation it applies to nothing and implies nothing.
Also, math doesn't claim to be "true." In the twentieth century a lot of really smart people used math to study math itself. And from a syntactic standpoint, the best you can do is agree that 1+1=2 follows from the Zermelo-Fraenkel axioms of set theory, with the usual von Neumann interpretation of the natural numbers.
But is 1+1 = 2 true? It is, about the world. But in math it's only true because of the way they define the symbols '1', '+', '=', and '2'. So this is tricky business, with the usual suspects having their say, from Russell to Frege, Wittgenstein, Godel, Turing, and all the deep thinkers about what it means to sling around strings of symbols according to rules.
As I understand it, I've stated the formalist position. But it seems to me that the math means nothing until you assign meanings to the symbols. It's true that math is inspired by physics. But math isn't physics.
What is mathematics?
Mathematics is a modeling language/toolbox.
What is a model?
A set of entities and relationships between those entities.
How do we come up with a model?
Through observations. We observe a system (eg: Universe), collect data and then figure out entities and relationships.
Why we need a model?
So we can "understand" the system
Can we create a model without observation?
Yes, in that case you would come up with entities and relations that are not result of observing some system (although the influence could be from various observations from various systems and various existing models), you can say those things doesn't exist in real world.
Any example of a model without observation?
The game of Chess and many other systems that we humans have designed that weren't already there in the real world.
Summary: Physics is about observing the universe we live in and try to create models of it using mathematics. You can use mathematics to create any model that describe a system which doesn't really exist aka potential universe.
So I actually really like this view on "The difference between math and physics is that physics describes our universe, while math describes any potential universe".
Let me try to explain why he said what he did: The idea is that if you pick a physics theory, for instance take the Standard Model, the theory is not all perfect, it describes the behaviour of the strong force, the weak force and the EM. But it doesn't do that just by itself, it needs a few free parameters. These parameters are specific to the world in which the laws of physics operate in. That I think is what he means by physics describes our universe.
Now if you talk specifically about the equations that describe phenomenon, they don't all use free parameters. You have to understand that a theory has to be comprehensive in it's explanatory scope, because of that it needs those free parameters. But the equations that describe something don't need the free parameters, so those equations can describe any universe, because when a theory uses them it can then in turn use different parameters.
I don't know any physics, but I've heard that a source of some anxiety among physicists is the possibility that the physical laws which obtain in our universe are just a kind of local accident, and that the constants of nature may vary outside of our Hubble volume. This hypothesis is offered as an explanation for fine tuning, for instance, and it has been given even more credibility (I'm told) by recent findings at the LHC. This possibility has always seemed to me like a pretty good argument for math as a career path.
I do worry that I am being too glib, however, and I hope any onlooking physicists will correct me.
The professor's statement is fairly accurate if you restrict physics to the "current known material universe. The current "real" universe is described as having 4 dimensions (3 spatial, 1 temporal). In contrast, with "pure" math, you can have an infinite number of universes, with an infinite number of dimensions - each! With the added advantage that one of these universes is the"real" universe that physics studies.
It is also possible that what your professor was trying to tell you, was that majoring in math would give you a "broader" base than majoring in physics. With a math base, you could easily continue into chemistry, biology, engineering, or any other science.
As with any short statement, it is possible to disagree with it. However, I think you get more benefit by being charitable and seeing what insight the fellow was talking about. I think his main point is that physics DOES pertain to our universe. Math is an abstract system.
This is a very cool insight. But it isn't necessarily a point in math's favor. There tends to be some loss of connection to the real world, when you move to the abstract. Physicists get to learn about electricity and things like that that we experience. Math majors don't. Similarly many biologists LOVE their topic because what could be more interesting than learning about our bodies. But if you want to learn the abstractions more power to you.
You asked for other writings and I was glad to see the V Arnold comments but surprised nobody mentioned this clip:
https://www.youtube.com/watch?v=obCjODeoLVw
In addition to the math versus physics, I would be attuned to where he mentions hypotheses based on bias. I.e. that it is OK to have them but realize it and don't be too prejudiced. It's a good philosophical point.
Finally in the context of what is a better major, I think that a lot of that comes down to YOU, not to which major is better. Is basketball "better" than wrestling. Maybe. You can make actually make arguments. Not definitive, sure, but you can find factors like team sport, notability, etc. to support a stance. However, what is best for YOU will likely have a lot to do with your physique, skills, aggressiveness, enjoyment, etc.
So...I definitely think that "this is better" field is a very low factor in your calculus. It should be more about what you enjoy, what you are good at. And also, I would be careful to consider what further training will be like and what jobs will be like. You might like freshman calculus and physics equally but find the move to proof mathematics a turnoff in the math curriculum (real analysis doesn't actually help you solve more integrals!)
Feynman had a negative reaction to math, his initial major at MIT, when he asked the professors what they did with higher math in later parts of math major and the answer was teaching more people to do that in the future. At first he switched all the way to EE. Then he decided that was too applied for him and settled on physics.
None of this is to argue you to physics (I'm even more applied than physics), because if you like math and are good at it, then fine. Also there are a few fields of math (real mathematicians will scoff) like statistics or operations research where you learn and perhaps even research fundamental math in the topic but there's also a pretty strong emphasis on collaboration with practitioners in the real world.
According to me he is right.
In physics we study the real universe and make concepts. That's why Physics describe our universe only.
But to represent any physics concept in theory we use math. And Math can represent both realistic, non realistic(virtual concepts). Means math can represent any virtual universe, which you can imagine. So you can say Math describe any possible universe.
Now let me explain, what I mean by virtual concepts:
In our(real) universe when you jump then you path will from ground to up then again down to ground => which can be represent by Parabola in math by using the mathematical value of gravity 'g'.
Now assume a universe in which if you jump then you go up then down, then up, then down => That also can be represent by math as Sin-Wave with value of 2 imaginary factor d -down force and u -up force.
I m not saying math is best. Its a different point. On one way, I'll say physics is best coz it is about our real life -> makes our life easy and more understandable, math may confused us. And in another way I can say math is best coz you can represent and process anything -> like if currently there is no physics concept here, even then we can process/calculate the concept and in near future we can invent a physics concept. "It should be everyone's personal opinion."
Mathematics is the genus and Physics is the species thereof. Mathematics does not need Physics but the latter does need the former.
Let's imagine that one of the dreams of some physicists was realized and we had an equation that somehow summed up answers to all relevant physics questions. Let's pretend this was like S + 2.34176447854...R = F/3. That's the heart of our universe.
But we could come up with infinitely many variations on this equation. A different irrational factor, different operations, etc. so if these were possible alternatives to the equation that we settled on, would they not be hearts of other possible universes?
What I believe your professor meant by saying "physics describes our universe while mathematics describes any potential universe" is that mathematics always studies a more general case of anything while physics just studies a special case of that anything. Your professor probably wanted you to focus on the word " any" rather than thinking whether or not there are any other universes existing. That's my point of view. To explain further, for example, Mathematicians have build theorems in n-dimensions but physicists are always interested in the special case, when n=3 or 4. When you will study mathematics you will talk about arbitrary forces for example but a physics student will just need you to give him a theorem about the gravity law. You as a mathematician wouldn't care whether or not the thing you are discussing exists in reality or is even possible or not but you'd still be able to proof it using logical reasoning and previous theorems and make your statement as general as possible. That's the beauty of mathematics that your professor was trying to sum up in just one sentence. P.S: This answer was inspired by Richard Feynman's lectures.
Math does not describe all possible worlds. It doesn't describe anything as such. It can be used as a net to throw over reality. When it is used to throw over the physical world it is shaped by features of this world which can be determined by experiment. Are these features mathematical in Nature? If you are a mathematical physicist probably yes. The physical world it is thrown on does show regular behavior. It would very unreasonable indeed if math wasn't effective in describing the world. How else could it be?
Of course there can be math to which there correspond no physical processes. And maybe there are other worlds corresponding to this math. We will never know. But that doesn't mean that math describess the world in the best and most fundamental way. It doesn't give a better understanding of the world than any other disciplin
Mathematicians maybe disagree with the last statement. Some of them see the world of math as the only world in existence. Like Plato saw the perfect world of ideas as the true world and the physical world around us as a dusty and stained shadow of that world. Mathematical structures are all that is real and contemplating them is to have a look in that pristine world.
Luckily we are not all mathematicians. We would be living in a pristine world but it would also be a terrifying world. Why? Because there would be no escape.
Your professor is right: The objects of study of physics and mathematics are completely different. Physics seeks to understand the universe, the reality we perceive, and because of this uses the scientific method. Mathematics, on the other hand, seeks to understand mathematical structures, and the method it uses to establish its truths is logical reasoning. So different are their objects of study that if a physical theory is known to conflict in serious ways with experimental results, it would be discarded in favor of a theory that conforms better with experiments (if one exists). In mathematics, there is not even an attempt to see whether the structures studies conform to reality (and the question itself is probably meaningless anyway).
Sidenote: One could ask about the parts of physics that don't deal with things we can experiment on (parts of string theory and so on). First, sometimes we can at least experiment on consequences of the theory. Second, those parts that truly have very little connection with reality would arguably fall more within mathematics.
