What is the Difference between Maths and Classical Mechanics?

[Fiery Graviton]

TL;DR

Our mechanics lecturer keeps repeating that we shouldn't think mathematically, I am trying to understand the "physicist intuition".

I am guessing this has been posted before if that's the case please inform me.

I was wondering the difference between mathematics and physics. What makes them so different yet so similar? Why does the mathematical intuition does not work on physics, or vise-versa. I know physics is less "deterministic" than mathematics. I am just curious as I want to become better in physics.

Thank you for taking your time to read.

 

[brainpushups]

There is a series of lectures that Feyman gave at Cornell which you may be interested in. The second lecture in the series is specifically about the relationship between math and physics. They are all worth a watch.

https://www.feynmanlectures.caltech.edu/messenger.html

 

[Herman Trivilino]

You need a conceptual understanding of both math and physics. Your professor is asking you to develop the latter without using only the former.

Here's why and here's the difference. Physics is the study of phenomena. The focus is on trying to understand how Nature behaves. It's not just a set of exercises in applied math, although it can tend to be thought of that way by students.

Your professor wants you focused on the phenomena. Physics is phenomenology.

It is not a set of math "word problems" where you search the wording of a problem for the algorithm you'll need to memorize your way to making answers.

Your professor wants you to be sense-making, not answer-making.

Here's an example. A ball is thrown upward at a speed of 4 m/s. How high will it go before it starts to descend?

Many students go equation hunting, looking for one that gives the answer. Your professor wants you thinking about how gravity affects the ball and consequently how that will affect the ball's motion. You'll end up solving for an answer the same way, but your thought processes will differ in a fundamental way.

This is a big problem for physics educators. How to get students away from thinking of physics as an exercise in applied math.

As the physics gets harder students who think of it only as an exercise in applied math get left in the dust. Your professor doesn't want that happening to you.

 

[QuarkyMeson]

I was wondering the difference between mathematics and physics.

Handwaving, basically.

I know physics is less "deterministic" than mathematics.

Anyway, what do you mean math is more deterministic than physics?

Why does the mathematical intuition does not work on physics, or vise-versa.

I'm struggling to envision what you mean here.

Your professor is probably asking you to look behind the mathematics and build physical intuition about what the problem is asking you to do. In your classical mechanics for example you can certainly learn say Lagrangian mechanics and learn what to do perfectly well mathematically without understanding what's going on physically or how you've managed to get there or what the big picture the problem is even asking is. You can even bypass understanding if you're looking at the problem locally using a d’Alembert approach or globally with least action. You're missing the physics. You're missing the interesting part of wondering, "why the **** does this work?" You're missing what would happen to your problem if I take your massless pulley and gave it mass. You're missing the physics.

Basically, he's likely saying "math good understanding better".

 

[Matterwave]

I was wondering the difference between mathematics and physics. What makes them so different yet so similar?

Physics heavily uses mathematics, but, at the end of the day, physics is about the real physical world and it's anchored by experiment. Physics, unlike math, is not an exercise in pure logic built from the ground up, in theorems and lemmas, on top of a set of axioms. At some point you need to make contact with the physical world.

To a physicist, mathematics is essential but not total. There is still a lot of physical intuition one builds that is not necessarily mathematical in nature.

For example, the statements "when I hit the breaks on this car, my body is going to lunge forward" or "when I'm free falling I feel weightless" are both physical facts building physical intuition without math. Of course you can (and should) use math to model those statements and make them precise, but the underlying physical intuition is very important.

At least this is how I view things.

 

[PeroK]

TL;DR: Our mechanics lecturer keeps repeating that we shouldn't think mathematically, I am trying to understand the "physicist intuition".

I am guessing this has been posted before if that's the case please inform me.

I was wondering the difference between mathematics and physics. What makes them so different yet so similar? Why does the mathematical intuition does not work on physics, or vise-versa. I know physics is less "deterministic" than mathematics. I am just curious as I want to become better in physics.

Thank you for taking your time to read.

I don't understand this type of thinking. It's a bit like asking what's the difference between chemistry and biology? Or, what's the difference between chili con carne and a cheeseburger? I can personally find no insight in comparing mathematics and physics. Instead, understand physics for what it is; and, understand mathematics for what it is.

Anyone who tells you not to think mathematically is a fool! You don't want to think ONLY mathematically. You may, if you wish, refer your mechanics lecturer to Roger Bacon:

Whoever then has the effrontery to study physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom.

Roger Bacon (1220-1292)

 

[javisot]

I was wondering the difference between mathematics and physics.

Physics=mathematics + natural philosophy

(Mathematical Principles of Natural Philosophy, Newton)

If you remove mathematics from physics, what remains is natural philosophy. If you remove natural philosophy from physics, what remains is abstract mathematics.

Using mathematical language to optimize natural philosophy and thereby explain certain aspects of reality opens up the possibility of conducting experiments, which are a fundamental pillar of physics.

 

[javisot]

TL;DR: Our mechanics lecturer keeps repeating that we shouldn't think mathematically, I am trying to understand the "physicist intuition".

I am guessing this has been posted before if that's the case please inform me.

I was wondering the difference between mathematics and physics. What makes them so different yet so similar? Why does the mathematical intuition does not work on physics, or vise-versa. I know physics is less "deterministic" than mathematics. I am just curious as I want to become better in physics.

Thank you for taking your time to read.

Agent?

 

[PhDeezNutz]

I gotta be honest.........Grad Classical Mechanics felt like a math class. Canonical Transformations, Hamilton-Jacobi Theory, Action-Angle Variables, etc. It felt completely divorced from physical intuition/reality (even though it is not, but I haven't reached that level yet to reconcile the two).

 

[Fiery Graviton]

Agent?

what's an agent?

 

[Fiery Graviton]

Also thank you for everyone's comment i have read them, I will check the feynmans classes as

https://www.feynmanlectures.caltech.edu/messenger.html

brainpushup has stated,

I do understand it little better but i guess i have to read more on natural philosophy. But again thank you for all your information

 

[martinbn]

Don't take this the wrong way, but did your lecturer really say that or is it something you think is the right thing to do, and you wrote that your lecture said it so that noone dismisses it of hand? If he did say it, what was the context (any specific examples, consepts, calculations...), and what was his exact wording?

 

[Fiery Graviton]

Don't take this the wrong way, but did your lecturer really say that or is it something you think is the right thing to do, and you wrote that your lecture said it so that noone dismisses it of hand? If he did say it, what was the context (any specific examples, consepts, calculations...), and what was his exact wording?

He said mathematics is how we understand physics however it's not pure mathematics, we have to develop physics intuition. What's right on physics might be completely wrong on mathematics and vise-versa.

 

[wrobel]

Speaking specifically of classical mechanics, there are plenty of complex, entirely counterintuitive problems that can only be understood through the analysis of differential equations, whereas physical intuition merely misleads you.

 

[wrobel]

On the other hand, once you see how a system behaves, you can always concoct a physical explanation for it. Predicting its behavior, however, turns out to be slightly trickier.

 

[Fiery Graviton]

On the other hand, once you see how a system behaves, you can always concoct a physical explanation for it. Predicting its behavior, however, turns out to be slightly trickier.

I think he was implying combining the mathematical frame work with the physical explanation, as he is Physical engineer normally.

 

[wrobel]

just an example

Euler top

 

[pinball1970]

He said mathematics is how we understand physics however it's not pure mathematics, we have to develop physics intuition. What's right on physics might be completely wrong on mathematics and vise-versa.

I always find this question interesting, I'm never sure how to answer the mathematics part.
What is it? Is there a consensus? Is it "philosophy?" A pf maths guy said that a while ago.

Physics is science and science is a method of exploring the universe and how it works via empiricism and theory.
Mechanics is canon balls and other projectiles so it's also observing and fitting those processes into theories with predictive power, I use this angle, this much gun powder/force and this weight, my ball will land over there.

Mathematics describes it but it doesn't have to, some maths just seems to be, exist, those deep relationships. None of it actually exists, X, e, I, dy/dx, π. An electron exists in the universe, a cell exists, CO2 exists but x does not, neither does 2.

EDIT: just to add, in this case I would say that mathematics is an abstract representation of what is going on with your projectile, What's going on is the physics. So gravity, air resistance, forces, velocity, the maths is F, dx/dt, dv/dt.

 

[erobz]

Mathematics describes it but it doesn't have to, some maths just seems to be, exist, those deep relationships. None of it actually exists, X, e, I, dy/dx, π. An electron exists in the universe, a cell exists, CO2 exists but x does not, neither does 2.

If it doesn't exist, how do you suppose it works so consistently on such deep levels? All of these things that you say "do exist", are also inescapably constructions of the mind. Can you observe the "physical universe" without a mind? A universe devoid of consciousness is no universe at all.

 

[Matterwave]

Discussion around whether numbers or math exists is in the domain of philosophy of math. The issue also involves the definitional point of what it means for something to "exist".

I think if you see a physicist say something along the lines of "numbers don't exist", you can generally read that as saying numbers don't exist in the physical world; numbers don't have the same existence as physical objects like tables or chairs; or the fundamental stuff that tables or chairs are made of is not the same as that of numbers. It might lead to less confusion. This is how I read such statements.

 

[symbolipoint]

TL;DR: Our mechanics lecturer keeps repeating that we shouldn't think mathematically, I am trying to understand the "physicist intuition".

I was wondering the difference between mathematics and physics. What makes them so different yet so similar? Why does the mathematical intuition does not work on physics, or vise-versa. I know physics is less "deterministic" than mathematics. I am just curious as I want to become better in physics.

Doubting very much that at this point my response necessary, but Mathematics uses Mathematics to explain and advance Mathematics. Classical Mechanics uses Mathematics to explain and advance Classical Mechanics (or the interchange among masses and energy).

 

[symbolipoint]

I can personally find no insight in comparing mathematics and physics. Instead, understand physics for what it is; and, understand mathematics for what it is.

Comparison alone does not help enough. One needs the contrast, too.

 

[erobz]

Discussion around whether numbers or math exists is in the domain of philosophy of math. The issue also involves the definitional point of what it means for something to "exist".

I think if you see a physicist say something along the lines of "numbers don't exist", you can generally read that as saying numbers don't exist in the physical world; numbers don't have the same existence as physical objects like tables or chairs; or the fundamental stuff that tables or chairs are made of is not the same as that of numbers. It might lead to less confusion. This is how I read such statements.

Since everything "found/examined/observed" in the physical world is mediated by the mind; At a fundamental level it actually is the case that what we "are experiencing as the physical world" is in fact just our internal mathematical constructs of it compiled from incomplete sensory input (a brain, trapped inside a meat sack equipped with a few key sensors for survival) - our internal mathematical structure is what we truly "observe - think about". It's all encompassing as our reality.

So I would argue, there is no valid argument for math/numbers being unreal, without what we experience as the "physical world" being sacrificed to the same notion.

 

[javisot]

I think if you see a physicist say something along the lines of "numbers don't exist", you can generally read that as saying numbers don't exist in the physical world; numbers don't have the same existence as physical objects like tables or chairs; or the fundamental stuff that tables or chairs are made of is not the same as that of numbers. It might lead to less confusion. This is how I read such statements.

In this sense, numbers exist just as a person's name exists. Among their functions, in addition to being abstractions of quantity, they also serve as simple labels.

 

[Matterwave]

In this sense, numbers exist just as a person's name exists. Among their functions, in addition to being abstractions of quantity, they also serve as simple labels.

This discussion will become philosophy of mathematics very quickly and that is above my paygrade. (Also I think rather off-topic... but forgive me for indulging)

But, at least one comment -- simple labels of what?

It seems to depend on what axioms you build your mathematical system on. If you start with the Peano axioms, for example, Axiom 1 gives you the symbol $0$ and then you may define symbols like $1\equiv S0$ or $2\equiv SS0$ etc. where $S$ is the successor function. You can build the natural numbers this way.

If you start with ZF (or ZFC) then you only have sets, and you'll need to make some different definitions based on how you encode (natural) numbers. For example, the von Neumann encoding, which is often given:

$$
\begin{aligned}
0 &\equiv \{\} \\
1 &\equiv \{\{\}\} = \{0\} \\
2 &\equiv \{\{\}, \{\{\}\}\} = \{0, 1\}
\end{aligned}
$$

But this is not the only possible encoding. I can't (and haven't) answer(ed) my own question.

 

[pinball1970]

This is how I read such statements.

That's what I meant. I can see two pennies and I can write two/2, I just did it, but I cannot go out into the universe and find 2. It is an abstract concept, it does not physically exist.

 

[pinball1970]

In this sense, numbers exist just as a person's name exists. Among their functions, in addition to

There is a difference, you exist and your name is Javisot, that is the name given to you and we do this with things also so we all know what we are talking about. 2 is a concept, it does not physically exist.

 

[pinball1970]

This discussion will become philosophy of mathematics

Apologies. However if we are going to compare mathematics with mechanics we need to know what those things are to begin with.
I do not know what mathematics is, I will follow the consensus, it it philosophy? I do not know enough about philosophy to consider it.
I know what maths can be used for, if it is moons orbits and forces and velocities in classical mechanics then it is very useful, essential in fact.

 

[pinball1970]

Doubting very much that at this point my response necessary, but Mathematics uses Mathematics to explain and advance Mathematics

Agree.

 

[javisot]

There is a difference, you exist and your name is Javisot, that is the name given to you and we do this with things also so we all know what we are talking about. 2 is a concept, it does not physically exist.

You can also call me 2 instead of javisot, if you prefer.