What is the Difference between Maths and Classical Mechanics?

[pinball1970]

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

Two is already taken, it represents an abstract concept that does not physically exist.

 

[javisot]

Two is already taken, it represents an abstract concept that does not physically exist.

I insist, call me 2.

https://en.wikipedia.org/wiki/Number, "A number is a mathematical object used to count, measure, order and label"

 

[Matterwave]

I do not know what mathematics is, I will follow the consensus, it it philosophy?

No, mathematics and philosophy are different topics. "Philosophy of Mathematics" is a subfield of philosophy which deals with issues involving the foundations of mathematics.

Rigorously defining mathematics is rather difficult. Perhaps the most useful thing to do here is just to keep in mind the operational definition as given, for example, in the wikipedia page: https://en.wikipedia.org/wiki/Mathematics

A rough taste of mathematics might be: given the rules of (a) logic (e.g. propositional logic or first order logic), and a set of (non-logical) axioms, what are all the statements we can prove from those axioms?

 

[PeroK]

Rigorously defining mathematics is rather difficult.

That's true, but we can and do introduce the concept of the infinite set of natural numbers without any philosophical difficulties. Even at the high school level, you are faced with this question. Does it makes sense that the natural numbers go on forever; or, do we have to stop at some point?

If you are mathematically minded, you come to the conclusion that this is perfectly valid. That these numbers themselves are an abstract concept that reflects the real world in some sense, but is not bound by physical limitations. And, we can imagine there is no largest number and work with that as a valid mathematical system.

Meanwhile, over in the English class, you are asked to read stories and even write a story yourself. And, most children are able to do this, and recognise fact from fiction. A philosopher might have a field day analysing in what sense a story exists: do the characters in a story exist? But, these philosophical question do not and should not prevent us from indulging in creative writing.

Meanwhile, in the sports class, children run around, oblivious to Zeno's paradox that makes motion impossible.

This is the key point: difficult and profound philosophical questions do not stop us doing things. Mathematics, writing, sports. We can do them regardless of any philosophical misgivings.

 

[Matterwave]

That's true, but we can and do introduce the concept of the infinite set of natural numbers without any philosophical difficulties. Even at the high school level, you are faced with this question. Does it makes sense that the natural numbers go on forever; or, do we have to stop at some point?

If you are mathematically minded, you come to the conclusion that this is perfectly valid.

Perfectly valid as consequences of (deductions from) axioms. In post #25 I gave two different axiomatic systems in which the natural numbers can be constructed or encoded.

In Peano Axioms, the infinitude of natural numbers arise from the axioms governing the successor function S:

For every natural number n, S(n) is a natural number. That is, the natural numbers are closed under S.

For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.

For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

You actually need all 3 to prove the infinitude of natural numbers (it's interesting to think through why).

In ZFC, there is the axiom of infinity: https://en.wikipedia.org/wiki/Axiom_of_infinity

This single axiom is enough to give the infinitude of the naturals.

This is all standard mathematics. It's not philosophical. But it's non-trivial. For example, that you need the axiom of infinity in ZFC and that it gives rise to the following fact:

[The axiom of infinity] guarantees the existence of at least one infinite set, namely a set containing the natural numbers.

Is non trivial. It's something I doubt school children, even if they are quite mathematically minded, would be able to argue for coherently.

That these numbers themselves are an abstract concept that reflects the real world in some sense, but is not bound by physical limitations.

I would argue that you are arguing philosophically here.

And, we can imagine there is no largest number and work with that as a valid mathematical system.

We can certainly imagine it. We choose axioms often because they give rise to structures that are useful or because they match intuition. From a historical perspective, axiomitization came later than many of the structures we are familiar with. People had been using natural numbers far before Peano or Zermelo and Frankel were ever born.

But, these philosophical question do not and should not prevent us from indulging in creative writing.

Definitely not. I agree with you. I hope you didn't read my posts as saying we should change the way we live our lives because philosophy is hard.

This is the key point: difficult and profound philosophical questions do not stop us doing things. Mathematics, writing, sports. We can do them regardless of any philosophical misgivings.

Yep, we can of course do things. But philosophy has its uses in elucidating the fundamentals and foundations.

The debate in this thread (around "do numbers exist?") seemed to be turning philosophical in nature and I was just pointing that out (and indulging myself by making some side comments).

 

[Dale]

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

The difference between math and physics is experiment.

 

[A.T.]

The difference between math and physics is experiment.

You can do repeated experiments to test a mathematical prediction of some probability. You can even test basic algebra by using sets of objects and counting.

But I think the role of experiments in math is very different from physics.

 

[erobz]

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.

For the skeptics on my post... When you open your eyes in the morning do you experience the universe as it is? No. You see a wall, you hear a coffee pot, you feel a crick in your neck, etc... all of the input available to our sensory system constructs the world around us as we become "conscious"...That internal construction of the world is NOT "the world", it is...a world model. The model is riddled with sensory holes - logical gaps - etc...it imperfectly rendered. Those gaps are filled by abstract thoughts. These thoughts exist in the same realm as our world model. Mathematics is not the structure of the outside world; it is the structure of what we experience "as the outside world"; It is the structure of our model. Mathematical laws in physics...also...world model - not world.

 

[PeroK]

Perfectly valid as consequences of (deductions from) axioms. In post #25 I gave two different axiomatic systems in which the natural numbers can be constructed or encoded.

In Peano Axioms, the infinitude of natural numbers arise from the axioms governing the successor function S:



You actually need all 3 to prove the infinitude of natural numbers (it's interesting to think through why).

In ZFC, there is the axiom of infinity: https://en.wikipedia.org/wiki/Axiom_of_infinity

This single axiom is enough to give the infinitude of the naturals.

This is all standard mathematics. It's not philosophical. But it's non-trivial. For example, that you need the axiom of infinity in ZFC and that it gives rise to the following fact:

Is non trivial. It's something I doubt school children, even if they are quite mathematically minded, would be able to argue for coherently.

To summarise: Children don't have to solve Zeno's paradox in order to run.

Likewise, we don't have to derive all mathematics rigorously before we can do it. Otherwise, there would be no maths before Russell and Whitehead. How did Euler do mathematics without ZFC?

Yep, we can of course do things. But philosophy has its uses in elucidating the fundamentals and foundations.

I don't think this is true. The foundations were worked out by logicians, using hard logic. Wittgenstein, I believe, worked on the foundations of maths, but he would never have solved the problem like Goedel.

Mathematicians, like Hilbert, were able to ask these question from purely mathematical considerations. They didn't need an overarching philosophy. In fact, the philosophising held them back, such as the partial rejection of Cantor's work.

 

[PeroK]

For the skeptics on my post... When you open your eyes in the morning do you experience the universe as it is? No. You see a wall, you hear a coffee pot, you feel a crick in your neck, etc... all of the input available to our sensory system constructs the world around us as we become "conscious"...That internal construction of the world is NOT "the world", it is...a world model. The model is riddled with sensory holes - logical gaps - etc...it imperfectly rendered. Those gaps are filled by abstract thoughts. These thoughts exist in the same realm as our world model. Mathematics is not the structure of the outside world; it is the structure of what we experience "as the outside world"; It is the structure of our model. Mathematical laws in physics...also...world model - not world.

We know about things beyond our senses. UV and IR light. Electrons, atoms, molecules; chemistry. We are able to go beyond our senses. Modern physics is not the physics of our senses. The quark model, for example.

Likewise, mathematics goes beyond anything that can possibly be mapped onto sense data. Mathematics, originally, was rooted in real world experiences. But, it has not been constrained by sense data for many centuries.

Moreover, pure logic is not based on sense data, or on real world experience. It's an abstract ability that we have to think beyond our experiences.

 

[Dale]

You can do repeated experiments to test a mathematical prediction of some probability. You can even test basic algebra by using sets of objects and counting.

But I think the role of experiments in math is very different from physics.

Yeah, I did think of that after I posted too. I should have said "physics experiments", but that just doesn't sound as pithy.

 

[erobz]

We know about things beyond our senses. UV and IR light. Electrons, atoms, molecules; chemistry. We are able to go beyond our senses. Modern physics is not the physics of our senses. The quark model, for example.

Likewise, mathematics goes beyond anything that can possibly be mapped onto sense data. Mathematics, originally, was rooted in real world experiences. But, it has not been constrained by sense data for many centuries.

Moreover, pure logic is not based on sense data, or on real world experience. It's an abstract ability that we have to think beyond our experiences.

You're missing my point. I'm not arguing the models are limited by what we can directly perceive by sensory input.

Let's look at UV. We don't directly sense it, but we deduced something was "there" in our world model. So we use our world model machinery to build instruments that interface with our biological sensors. The result is a detection via a blip on a screen or whatever. Then we update the internal world model for "electromagnetic waves" via inference.

Extending our knowledge of the world model doesn't bypass the world model; it extends it or refines it.

 

[Matterwave]

To summarise: Children don't have to solve Zeno's paradox in order to run. Likewise, we don't have to derive all mathematics rigorously before we can do it. Otherwise, there would be no maths before Russell and Whitehead. How did Euler do mathematics without ZFC?

Ok. I'm not sure why you are specifying this again to me. Do you think I don't agree with this?

Or do you have an issue with axiomatic math (or ZFC or Peano axioms) in general?

I don't think this is true. The foundations were worked out by logicians, using hard logic. Wittgenstein, I believe, worked on the foundations of maths, but he would never have solved the problem like Goedel.

To be very clear, the statement you responded to is:

Yep, we can of course do things. But philosophy has its uses in elucidating the fundamentals and foundations.

So your contention is philosophy has no use in elucidating fundamentals and foundations, is that right?

If it is your goal here is simply that I stop mentioning philosophy, sure I can do that. I already mentioned I think it's off topic.

Mathematicians, like Hilbert, were able to ask these question from purely mathematical considerations. They didn't need an overarching philosophy. In fact, the philosophising held them back, such as the partial rejection of Cantor's work.

Ok, again, I'm not sure why you keep mentioning this to me.

My statements have remained rather narrow. Specifically, I did not say:

1. You need to first axiomatize math to "do math".
2. Only an overarching philosophy can provide a foundation.
3. Philosophy of math is more important than Logic.

Or at least I don't think I said any of this.

 

[PeroK]

You're missing my point. I'm not arguing the models are limited by what we can directly perceive by sensory input.

Let's look at UV. We don't directly sense it, but we deduced something was "there" in our world model. So we use our world model machinery to build instruments that interface with our biological sensors. The result is a detection via a blip on a screen or whatever. Then we update the internal world model for "electromagnetic waves" via inference.

Extending our knowledge of the world model doesn't bypass the world model; it extends it or refines it.

Yes, but the blip on the screen is just a blip. It could mean anything. There is nothing inherent in a blip to distinguish one phenomenon from another. It might be a heartbeat; it might be a muon detection; it might be an increase in a share price on the stock market.

An animal is limited by its senses. We are not. We can understand things that we cannot directly sense. Stock market prices are, perhaps, a good example. They have to be rendered into sense data; but, the prices themselves are not sense data.

PS an animal with our senses cannot perceive the stock market. It can perceive the blip on the screen, but it cannot map that to something that is itself abstracted from sense data.

 

[PeroK]

Ok. I'm not sure why you are specifying this again to me. Do you think I don't agree with this?

Or do you have an issue with axiomatic math (or ZFC or Peano axioms) in general?


To be very clear, the statement you responded to is:

So your contention is philosophy has no use in elucidating fundamentals and foundations, is that right?

If it is your goal here is simply that I stop mentioning philosophy, sure I can do that. I already mentioned I think it's off topic.


Ok, again, I'm not sure why you keep mentioning this to me.

My statements have remained rather narrow. Specifically, I did not say:

1. You need to first axiomatize math to "do math".
2. Only an overarching philosophy can provide a foundation.
3. Philosophy of math is more important than Logic.

Or at least I don't think I said any of this.

I apologise if I misunderstood what you were saying.

 

[Matterwave]

I apologise if I misunderstood what you were saying.

No worries at all!

I always felt like we were just chatting (and not arguing) and I like to chat about many random topics in math, physics, logic (and you can guess Philosophy haha, though I try to keep that to a minimum here). :)

I was hoping that my readings in philosophy and logic would help to make my points clearer, but I still need to work on this. That's on me.

 

[Herman Trivilino]

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.

But once you've done that analysis you have the opportunity to develop an understanding that makes those problems intuitive.

I think that's the point of the OP's professor's advice. Start developing the physical intuition during the introductory courses when it's easier. Waiting until the upper-level courses leaves you unable to develop an intuition. You're stuck doing math exercises instead of phenomenology. That is what I meant in Post #3 by being left in the dust.

 

[erobz]

Yes, but the blip on the screen is just a blip. It could mean anything. There is nothing inherent in a blip to distinguish one phenomenon from another.

Exactly my point. There is nothing inherent in a blip that makes it distinguishable as a particular phenomenon...it becomes distinguishable as a particular phenomenon via the current world model you are using.

 

[pinball1970]

As an afterthought.

If you would have asked that question when I was an A level student I would have answered differently and more simply.
Classical mechanics is part of Applied mathematics and just "Mathematics" I would have said was "Pure mathematics."
That is what my books said, Bostock and Chandler 1983.
The situation has become more interesting since then.

 

[Matterwave]

Classical mechanics is part of Applied mathematics and just "Mathematics" I would have said was "Pure mathematics."
That is what my books said, Bostock and Chandler 1983.

Do you maybe have a quote? This seems just wrong? Applied mathematics sources some topics from Classical Mechanics (e.g. symplectic manifolds arise naturally in Hamiltonian dynamics) but to say Classical Mechanics is "part of" applied math seems to ignore the entire empirical side of things.

 

[berkeman]

This thread has veered off of a technical course, and is closed for Moderation.