Mathematics proof vs Physicists proof

[farleyknight]

Firstly, a little background: I'm a compsci major with a new found love of mathematics and particularly formal logic. To put it bluntly, I like symbols. Reality doesn't really give me a warm fuzzy feeling inside, but when I see that two algebraic equations are identical, I feel like god is smiling on me (and, as you would guess, when I make a mistake, it's like I'm being tortured by demons)

Anyways.. How do physicists prove theoretical / abstract concepts? I understand that all principles must be held up to the light of physical evidence and measurement, but say that you don't have that. Say you're having a conversation with a friend, a thought experiment or what-have-you. You're just trying to convince them that what you say is true, but without the benefit of experiment. How do you do this without the use of formal logic like in mathematics?

Is convincing another physicist more about experience with a long list of principles, gathered by old dudes over centuries, and just becoming familiar with them, or is there a way to derive secondary laws from the primary ones?

 

[Lsos]

The problem is that there is no "proof" in science. Proof is, as you seem to suspect, a mathematical concept. So if you are having a conversation with a friend and are trying to convince them of something, you will need either experimental and observational evidence, as you pointed out, or math. Preferably both. And the more you have of these than the closer you can get to this "proof"...although, you can never truly attain it.

If you go and try to convince someone of some theory without either, then you are probably a crackpot. How much of a crackpot is directly proportional to how extravagant your theory is, and how little evidence you have for it.

However, you can definitely derive new theories from existing ones. For example, I can use the theory of gravity to derive the theory that if I drop a bowling ball on someone's head, they will get hurt/ die. I don't need to make a direct experiment to be fairly certain of this.

 

[wisvuze]

you can obviously mathematically or logically derive certain things that must hold if your original findings were true. For example, if you know that if the velocity of an object is porportional to its motion with respect to time, then its motion cannot follow the equation ct^2. But the foundations of physics will always be observational and experimental.

An interesting thought is what physicists are actually looking for, I wouldn't say that a physicist actually wants to find what is "true" about the universe, a physicist only wants to come up with a useful system with which they may view the world. "An apple doesn't fall down because of gravity, it falls down because it falls". On the other hand, finding "absolute truths" in the universe is a very romantic and directionless goal. Many things are true, you can say something that must be true - and this is how arbitrary it can be.
I originally wanted to study math - and while doing math is fun, physics is just so much more exciting.

 

[nicksauce]

There is still lots of mathematical proofs in physics. Look at a GR book like Wald, or Hawking&Ellis. In there you will see lots of Theorem/Proof blocks.

 

[farleyknight]

There is still lots of mathematical proofs in physics. Look at a GR book like Wald, or Hawking&Ellis. In there you will see lots of Theorem/Proof blocks.

Hmm.. Turns out I have a pdf of Wald.. Looking at it now.. Seems like there are a couple of proofs.. I'm in Calc III now, so I recognize but am not familiar with the operators they use, but that's only a matter of time.. I have no real understanding of what a tensor other than that it was described as an n-dimensional matrix. At the very least I don't feel like I'm being talked down to and asked to use my intuition for everything like my undergrad physics book does. I hate that crap... I'll add it to my wishlist.. Maybe I'll pick up a copy when I have the money.

 

[nicksauce]

Look at Chapter 8 of Wald especially. It reads like a math textbook.

 

[farleyknight]

Look at Chapter 8 of Wald especially. It reads like a math textbook.

Ahh... okay, I see what you mean.. This is probably the kind of physics that people fall in love with :) Not the "A train travels east at 100 km/h while a car travels north at..." kind of crap that I was exposed to previously...

 

[ChmDudeCB]

The proof is in the pudding.

We have a few sets of fundamental functions, postulates, constants, and operators from which we can derive many equations which we use to calculate many things...and it works.

For instance, Newton's equations can be used to derive the ideal gas equation with a thought experiment of how ideal gasses work (spheres of mass bumping into each other in free space). The ideal gas equation is experimentally found to work with near ideal gasses with in ideal ranges. We have corrections for non ideal behavior that comes from the electromagnetic interactions and the fact that atoms take up space. These electromagnetic interactions are, fundamentally, calculated from quantum mechanics and electromagnetic force equations. When we apply these corrections the math nearly identically describes measured values. Quantum and electromagnetic force equations were both developed by different people of different fields at different times and are independently verified experimentally.

If you looked at my Physical Chemistry book (basically a class on the molecular approach to quantum, thermo, and stat mech) you'd think it were a math book with real world examples. In fact, between each chapter is an appendix on things like operators and complex numbers and what not.

Edit:
And for that matter, many contributions to physics have come from mathematicians. Newton developed "modern" calculus.

 

[Andy Resnick]

<snip>
Anyways.. How do physicists prove theoretical / abstract concepts? I understand that all principles must be held up to the light of physical evidence and measurement, but say that you don't have that. Say you're having a conversation with a friend, a thought experiment or what-have-you. You're just trying to convince them that what you say is true, but without the benefit of experiment. How do you do this without the use of formal logic like in mathematics?
<snip>

Physicists (in general) are notorious for the lack of mathematical rigor. What constitutes a 'theorem' in physics is usually a half-baked derivation that mathematicians laugh at. Which is fine: Physics deals with the real world, not an abstract human invention.

If you would like to see a couple of mathematicians present an axiomatized branch of physics, check out "Mathematical Foundations of Elasticity" by Marsden and Hughes (Dover). I guarantee you that it looks like no physics book you have ever seen.

 

[farleyknight]

Physicists (in general) are notorious for the lack of mathematical rigor. What constitutes a 'theorem' in physics is usually a half-baked derivation that mathematicians laugh at. Which is fine: Physics deals with the real world, not an abstract human invention.

If you would like to see a couple of mathematicians present an axiomatized branch of physics, check out "Mathematical Foundations of Elasticity" by Marsden and Hughes (Dover). I guarantee you that it looks like no physics book you have ever seen.

Hmm.. 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?

 

[pgardn]

Hmm.. 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?

I suggest you never ever read books on evolution. Science sometimes has a very practical but messy side. We make models that work well, but don't conform to great mathematical rigor because you can't use math, the phenomena you are studying is way too complex. Does not mean its a waste.

When you come up with a mathmatical formula that tells us exactly what time in a persons life a cancer cell will go ballistic I will bow to you. There are very concrete differences between science and math. And I love math, but am also skeptical of some basic foundations. Kurt Goedel is quite interesting imo.

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...

 

[farleyknight]

I suggest you never ever read books on evolution. Science sometimes has a very practical but messy side. We make models that work well, but don't conform to great mathematical rigor because you can't use math, the phenomena you are studying is way too complex. Does not mean its a waste.

Erg.. must resist urge to start debate.. erg..

Well, much of evolution has not been formalized but that doesn't mean it cannot be. Gregory Chaitin proposed that a field of mathematics called metabiology be developed and studied that would do exactly that. No such field exists but it would be a place where Darwin's theory of evolution could be "proved", given sufficient conditions.

When you come up with a mathmatical formula that tells us exactly what time in a persons life a cancer cell will go ballistic I will bow to you. There are very concrete differences between science and math. And I love math, but am also skeptical of some basic foundations. Kurt Goedel is quite interesting imo.

And incidentally, Kurt Godel was probably the first lisp programmer :)

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?

http://xkcd.com/435/" is my general feeling about the topic. I hope I don't anger too many with it :)

 

[ChmDudeCB]

When you come up with a mathmatical formula that tells us exactly what time in a persons life a cancer cell will go ballistic I will bow to you. There are very concrete differences between science and math. And I love math, but am also skeptical of some basic foundations. Kurt Goedel is quite interesting imo.

Once we fully understand the mechanism of cancer we will be able to.

Mathematics is used to describe physics which dictates chemistry which dictates biology.

 

[farleyknight]

So now I really want to know.. Since I'm browsing Amazon for the topic: does there exist any books that can do this level of axiomatization for undergrad physics?

 

[Bob S]

How do physicists prove theoretical / abstract concepts? I understand that all principles must be held up to the light of physical evidence and measurement, but say that you don't have that. Say you're having a conversation with a friend, a thought experiment or what-have-you. You're just trying to convince them that what you say is true, but without the benefit of experiment.

Physicists can and do find exact solutions to basic problems without doing an experiment. Physicists would never have found the exact fastest path of a frictionless bead sliding down a curved wire from point A to point B by experimentation. One physicist, Isaac Newton, calculated the correct curve in 1697. See brachistochrone solution and animation in

http://curvebank.calstatela.edu/brach/brach.htm

http://mathworld.wolfram.com/BrachistochroneProblem.html

Bob S

 

[pgardn]

Once we fully understand the mechanism of cancer we will be able to.

Mathematics is used to describe physics which dictates chemistry which dictates biology.

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? The complexity piles up on the probability to make this way too difficult.
When will we see a unified field theory? Since we start with the most basic science.

 

[pgardn]

maybe we should just try and get the weather right...

I watched the Jetsons, no rocket cars yet. Very dissappointing.

 

[pgardn]

Well, much of evolution has not been formalized but that doesn't mean it cannot be. Gregory Chaitin proposed that a field of mathematics called metabiology be developed and studied that would do exactly that. No such field exists but it would be a place where Darwin's theory of evolution could be "proved", given sufficient conditions.

I would not expect evolution to be proved. Its the mechanisms of evolution that we attempt to quantify with things like population genetics. I would expect that we might be able to predict conditions at a later time. This is basically what most physics does. And I don't personally see this happening in this area. I see attempts, but it will be jelly because the variables are out of control.

 

[dx]

Hmm.. 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?

It's always good to be precise and think clearly, but the kind of axiomatic approach used by mathematicians is rarely useful (or even possible) in physics.

 

[farleyknight]

I would not expect evolution to be proved. Its the mechanisms of evolution that we attempt to quantify with things like population genetics. I would expect that we might be able to predict conditions at a later time. This is basically what most physics does. And I don't personally see this happening in this area. I see attempts, but it will be jelly because the variables are out of control.

You wouldn't be able to predict much of anything. That's not quite the point. 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. What would be the immediate benefit of this? Well, you may be already aware of this, but the field of genetic algorithms would benefit most certainly. Perhaps bioinformatics or quantitative biology? Who knows.. But a mathematical proof would be nice to finally kill of any notion of intelligent design.

 

[farleyknight]

It's always good to be precise and think clearly, but the kind of axiomatic approach used by mathematicians is rarely useful (or even possible) in physics.

It's useful because it helps me to understand it clearly :) A good general / abstract proof is not just an axiomatization but a very thorough and complete explanation. It carries out the very special cases. It does not bother giving actual examples (although they may be illustrative, they are not the proof itself). And most important, it does not rely on intuition. If I'm taking a test and I give some crazy answer, and I tell the teacher "but it's right because I used my intuition!", he'll probably fail me on the spot. But that's what I'm expected to use to understand certain physics principles in my undergrad book. Screw that. If it cannot be proved, take it as an axiom. But if it can, I'd like to see it.

 

[pgardn]

You wouldn't be able to predict much of anything. That's not quite the point. 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. What would be the immediate benefit of this? Well, you may be already aware of this, but the field of genetic algorithms would benefit most certainly. Perhaps bioinformatics or quantitative biology? Who knows.. But a mathematical proof would be nice to finally kill of any notion of intelligent design.

A mathematical proof would do nothing to stop these ideas. You don't utilize reasoning with people that did not use reason to establish their position in the first place.

The ultimate reason for using math in science is to be able to predict future conditions. Or look back into the past for a possible glimpse of how we got to present status. My take...

 

[gmax137]

A mathematical proof would do nothing to stop these ideas.

That's right.

I don't really understand the OP's issue. Physics is about the real world, a.k.a. "our universe." How can you "prove" anything relevant to that? After all, this (all that you see & feel) could all be a dream of some butterfly. To make progress, we choose to discount such notions. But we don't delude ourselves that we can construct 'proofs' regarding how the world works. And even if you do, someone will come along and find the faults in your proof. Didn't Kant & company 'prove' that the world has three dimensions?

 

[andrewbb]

IMO, be careful of mathematical proofs. Mathematics is a symbolic representation of reality. Not reality itself. It is a form of language and can be used to communicate, but never expresses the actual reality.

EG. Zero is not a number.

 

[farleyknight]

That's right.

I don't really understand the OP's issue. Physics is about the real world, a.k.a. "our universe." How can you "prove" anything relevant to that? After all, this (all that you see & feel) could all be a dream of some butterfly. To make progress, we choose to discount such notions. But we don't delude ourselves that we can construct 'proofs' regarding how the world works. And even if you do, someone will come along and find the faults in your proof. Didn't Kant & company 'prove' that the world has three dimensions?

I can let go of the idea that certain things cannot be mathematically proved but are still evident by observation. My real problem is when a teacher asks me to solve a problem and I cannot give him a rigorous explanation of why the result goes such-and-such a way due to the lack of rigor of the principles of physics.

 

[Hurkyl]

EG. Zero is not a number.

 

[farleyknight]

I'm guessing he means that it's not a number because it cannot be expressed in terms of countable objects in the real world..

Records show that the ancient Greeks seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.
​

http://en.wikipedia.org/wiki/0_(number)#History_of_zero

 

[Andy Resnick]

<snip>
My real problem is when a teacher asks me to solve a problem and I cannot give him a rigorous explanation of why the result goes such-and-such a way due to the lack of rigor of the principles of physics.

I don't understand what you mean- can you give a specific example?

 

[farleyknight]

I don't understand what you mean- can you give a specific example?

Hmm.. damn.. I'm going to have a hard time coming up with an example, even though I had this feeling quite a lot while studying last semester.. 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!"..

In other words, I felt as though I needed a formal proof that such a quantity don't matter. Otherwise they could affect the answer. My math teacher would never let me get away with such nonsense. I remember staring at the test for like 10 or 20 minutes trying to convince myself that it was okay but regardless of this, I couldn't.. My rational mind must have blanked out at that point because when I saw it returned to me none of the answers made any sense..

There are likely more.. If I dig out my physics book I could probably find some, I'm sure.

 

[andrewbb]

I don't understand what you mean- can you give a specific example?

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.

 

[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.