Mathematics proof vs Physicists proof

[farleyknight]

My example would be "kinetic energy". That is a loose, general phrase that is almost referred to as a discrete substance when the underlying concepts are insufficiently explored. The mathematics of it describe a relationship but not the actual.

Ah yeah, kinetic and potential energy both gave me headaches.. My mind would start wandering and trying to come up with counter-examples and paradoxes where the definition just didn't make sense or the rule was broken. I managed to curb those thoughts for the most part and just focus on the problems at hand which luckily didn't test my very naive assumptions. But I still needed help from the people on this forum to get past all that..

 

[Andy Resnick]

<snip>The first one that comes to mind, and something that I kinda got over so it's no longer an issue, although it still feels funny, is when one says in physics that a quantity is "too small to be considered and can be neglected".. It came up a couple of times during lectures but I paid the idea no mind. It didn't seem applicable. But when it showed up in a test, my very formalist mindset said "But WHY? Just give me the number and I'll decide for myself!"..

<snip>

A lot of problems in physics are linearized in order to generate an analytic solution. Sometimes, the linearization fails: a common example in mechanics is when the solution bifurcates. However, if the linearization process works (and there are formal theorems which determine this), then the solution to the full problem can be expanded as a Taylor series. This is sometimes called a 'perturbation expansion'. Truncating the series as an approximate solution is what is meant by 'small enough to be neglected'.

 

[andrewbb]

"Heat" is another of those abstract concepts that some physicists almost refer to as a discrete substance. IMO, it is a property of the fluid and should be discussed as temperature and its transferrence.

 

[farleyknight]

A lot of problems in physics are linearized in order to generate an analytic solution. Sometimes, the linearization fails: a common example in mechanics is when the solution bifurcates. However, if the linearization process works (and there are formal theorems which determine this), then the solution to the full problem can be expanded as a Taylor series. This is sometimes called a 'perturbation expansion'. Truncating the series as an approximate solution is what is meant by 'small enough to be neglected'.

Ah yeah, I can see dynamical systems theory having a lot of applications in physics, especially at very large or very small distances.. I managed to at least half-convince myself that leaving out small values wouldn't affect the final based on very rough draft proof using significant figures, i.e. a subset of the rational numbers which have fixed length representation. Given some set of numbers represented in such a format, it would be possible to prove certain properties that it would never change the final result based on their relative sizes.. I doubt it's entirely correct but at least I could sleep at night..

 

[pgardn]

Energy is a very abstract idea imo. But making attempts to identify what it means in certain situations has been very useful. Thats really what it all comes down to for many of us (humans). Sometimes we want to find out the way things seem to work so we can say "so..., that's the way it is" and it feels comfortable because someone else comes along and utilizes the purer idea for some technological purpose that allows us to control our environment to suit our desires.

But deep down I think most realize being useful, and at the same time conforming to some mathematical principals, don't necessarily marry well. And then the quantum stuff where the math seems to be driving the search (experiments) seems very bizarre. It makes sense in my mind because its just modeling. I just think that sometimes the actually symbolic naming of some of this stuff "strings" leads to metaphorical usage that lead people to wrong-headed notions. But we, as humans, rely so heavily on metaphor and analogy, its understandable even though it is limiting, and flat out wrong sometimes, we are indeed satisfying our strong desire to control our world. Taking a Platonic type of view that we should not clutter our ability to reason by using our senses to observe phenomena is an extremely cold way of viewing what type of animals we really are. But possibly necessary in certain situations.
I promise you I will slap a fire ant crawling on my leg. And I will use various chemical tricks to keep them from biting and stinging me. And others will try to understand this beast for the sole purpose of stopping it from stinging and biting. And the math they use to understand this complex social animal will not properly adhere to all mathematical rigor because its a very complex situation.
But I digress...

 

[sponsoredwalk]

Quite an interesting thread, even though I should be studying I got quite a few ideas reading this thread & I think I have something useful to contribute to the conversation.

Why can't Halliday and Resnick talk like that? Geez.. It's like, how does it make you feel when someone tries to explain centripetal acceleration by talking about what it feels like to be on a ferris wheel, something I probably haven't done for years, instead of just giving me the god damn theorem in R^2 like I'm an adult?

If I didn't stop using Halliday and Resnick when I did I would have quit studying physics, what a terrible book! Jeesh, Pure terrible... But, talking about how it "feels" to be on a ferris wheel doesn't make a book any less "adultey", it's just the quality of that particular book...

If you want more "axiomatic" undergrad physics I suggest you get Kleppner & Kolenkow's "An Introduction to Mechanics". It's short & is jam packed with a lot of math and very little use of numerical example. There's also the book Introduction to Classical Mechanics by David Morn that is pretty hefty, especially compared to Halliday/Resnick, [in the undergrad sense].

As for the rigour you seek, My advice, [for what it's worth], is to relax because you've still got a lot of it coming to you, (you mentiond you're in Calc III and are using Halliday so I assume that level). There are some things that really annoy me for their vaguery but you have to A) deal with it remembering to remember that you still have unanswered questions on a topic, or B) Look elsewhere for an explanation. I practise both lol.

If you're looking for serious elementary rigour, I can suggest something that I haven't studied but looks very interesting to me for when I have time in the future.

Do you know who Michael Spivak is? Well he has written an [as of yet] unfinished course on Physics for Mathematicians, "Elementary mechanics from a mathematician's viewpoint". Check here, http://en.wikipedia.org/wiki/Michael_Spivak, to get the pdf and the video's if you're interested, based on the quality of his Calculus book I'm dead set on reading this pdf in the future, no buts...

As for Kinetic & Potential energy, these topics seem very vague (but still interesting) to me too, you just have to accept them for now. What I am aware of is how important & exact they become when looking at the more mathematical Lagrangian/Hamiltonian formulation of Classical mechanics. I think you'll find a few answers when you get there, (I assume & don't know if you know this stuff already lol..). Hey, if you're not happy then ask questions on these forums

There are mathematical philosophers that tear up quite a few assumptions made by mathematics. We are only human... You think a hominid evolved over 100,000 years that somehow came up with a symbolic language is going to reach some sort of math nervana?

stay humble...

Nice it's been around 2000 years since we began, who knows in another 2000

When you go up a level to chemistry you increase the complexity enormously. And then take a step up to Biology... its nuts. The math becomes less rigorous because the complexity in modeling is crazy hard.

What breakthroughs do you see in biology and physics that will EVER allow us to predict when a person will die to the very second right when he is born?

My understanding of Quantum physics [quite basic] tells me that this is theoretically impossible. Maybe I'm wrong but due to the nature of electrons & the randomness they would almost certainly act in a way that would change your predictions. I'm not sure, I'll get back to you in about 4 years

The point would simply be to have a paper-and-pencil simulation of evolution that would show that given certain properties of an organism (or whatever mathematical formalism we use instead) it would "mutate" and "evolve" towards a set of fitness constraints.

Even things like wavelengths of light etc... can alter base pair orderings & damage parts of dna etc... If you want to find a mathematical formalism you can start by using a simple combinatorics equation to see how much complexity you'd need to account.

Holding every single other factor constant, [environment, cellular & atomic activity etc...],
Count every single base pair in a "standard" sample of dna, compute the possibility of what happens when the first base pair is changed to one of the three other possibilities. This is the number of potential experiments you do to find out what dna's full theoretical capability is. Then, Holding every single other factor constant, [environment, cellular & atomic activity], take note of the macroscopic outcome of what happens on 3 test subjects who all have a different base pair to that of your "control" subject throughout somebodies life [with every single other base pair remaining stable]. Then, redo it all with a separate test subject with this base pair missing.

Once you've collected that data, move on to the second base pair. Rinse & repeat until you go through every single base pair in this particular dna sequence.

Then, do it all again by changing the first & second base pairs & acounting for them being missing. then with three base pairs etc...

You've successfully predicted what dna does!

But that is all about DNA alone! Dna is not the only factor determining evolution... Add up all the proteins & pretty much everything that goes on throughout somebodies life... Nevermind atoms, intermolecular bonds, random Van der Walls fluctuations etc... Need I continue...?

[That was a word problem, you'll have to convert that into symbols yourself ]

got a bit carried away... Back to the books

 

[farleyknight]

If you want more "axiomatic" undergrad physics I suggest you get Kleppner & Kolenkow's "An Introduction to Mechanics". It's short & is jam packed with a lot of math and very little use of numerical example. There's also the book Introduction to Classical Mechanics by David Morn that is pretty hefty, especially compared to Halliday/Resnick, [in the undergrad sense].

Do you know who Michael Spivak is? Well he has written an [as of yet] unfinished course on Physics for Mathematicians, "Elementary mechanics from a mathematician's viewpoint". Check here, http://en.wikipedia.org/wiki/Michael_Spivak, to get the pdf and the video's if you're interested, based on the quality of his Calculus book I'm dead set on reading this pdf in the future, no buts...

Thanks for the recommendations! I will definition check out the books you mentioned.. It's funny that you mention Spivak. I didn't know he did work in Physics. I am pretty familiar with his Calculus textbook. I would much rather be working with books like that than the typical stuff you get as an undergrad. At about the time I was first learning calculus I was also first doing proofs, although I still didn't grasp the vernacular entirely. I was still using inference arrows and trying to boil plain-English proofs into nuts & bolts symbols. I vaguely recall having a tough time in calculus but I wasn't really sure why, even though I was doing fairly well in discrete math. My scores starting getting lower and eventually I just dropped calculus. I figured that if I was going to take it again I might start with something that would really force me to learn..

Around that time I found Spivak's Calculus and I felt it was filling in the questions that I didn't have the words to ask, like why the limit of 1/x = infinity as x approaches 0. I saw stuff like that in Stewart's and although I could somewhat take his word for it, I still felt like I didn't know what was going on. When Spivak asked that I prove such identities, I started to really understand how calculus worked. From there I figured I'd prove the various derivative properties and so on.. I only stopped working with it because I was too busy with other courses and learning the "practical" stuff. At some point I want to come back to it and see how quickly I can work through it.

 

[farleyknight]

Check here, http://en.wikipedia.org/wiki/Michael_Spivak, to get the pdf and the video's if you're interested, based on the quality of his Calculus book I'm dead set on reading this pdf in the future, no buts...

First we have to understand rigid bodies", since, after all, the rigidity
of the lever is an absolute necessity for it to work, and if one hasn't already
analyzed rigid bodies, then one simply isn't in a position to give an explanation
of the lever.

Haha! Spivack FTW!

 

[pgardn]

But that is all about DNA alone! Dna is not the only factor determining evolution... Add up all the proteins & pretty much everything that goes on throughout somebodies life... Nevermind atoms, intermolecular bonds, random Van der Walls fluctuations etc... Need I continue...?

[That was a word problem, you'll have to convert that into symbols yourself ]

got a bit carried away... Back to the books

This reminds me of when I first read about cloning. (I know we are discussing shifts in populations with evolution). There were some people who said they wanted to be cloned so they could live longer. I was stunned. They thought an exact (really inexact) copy of their initial DNA would be subject to exactly the same set of environmental circumstances that would interact with this "identical" DNA and this would in fact be them?
I was just floored.

All these different Physcis texts are designed for different audiences. The Halliday/Resnick book you speak of I suspect (actually I know) is used for AP-B Physics in many High schools. Physics for non-math/physics majors.