Physical Intuition, Not Mathematics (2011) • Hacker News Board

Re: Physical Intuition, Not Mathematics (2011)

#31

Post by tjradcliffe » Sun, Jul 05, 2015, 5:07 AM UTC

gh02t wrote: post #16

newsposter123 wrote: post #2

I would be interested to know more as to how people think in the language of mathematics when deriving and reading mathematical formulas as I really struggle with it if I don't get the intuition.

Well, as a working mathematician I find that the hard part is typically coming up with the right definitions. Once you have the right concepts clearly defined (in terms of things you already know), the actual new ideas tend to arise with ease. Similarly, it is often the case that new fields of mathematics arise from someone defining a new concept that was previously imprecise. Once you have the right language to disc…

Mathematicians use and think about math in a way that is almost completely orthogonal to the way physicists use and think about it. To physicists math is more like a natural language, and definitions are conveniences. This is why mathematicians often say physicists can't do math: by their definition of math this is correct.

For physicists the math becomes a language and a safety net and a set of heuristics that let us simply the problem to the point of being about to reason about it effectively. A great deal of what we use math for amounts to book-keeping. The human imagination is as capable of dreaming up impossibilities as it is incapable of dreaming up the way the universe actually is, and math helps us avoid doing the former while we use systematic observation, controlled experiment and Bayesian inference to figure out the latter.

Because "thinking about the mathematical representation of physical reality" is such a profoundly unnatural, unintuitive act, and because the math is so strict and simple, it is very hard to for us to use it to imagine impossibilities, whereas if you have a conversation with a layperson you will find they almost instantly run off the rails into nonsense because they don't have the math to keep them on track. So laypeople believe in perpetual motion machines and the like with surprising ease, because they "just make sense" to their intuition (which maps pretty well to Aristotle's physics).

Re: Physical Intuition, Not Mathematics (2011)

#32

Post by tel » Sun, Jul 05, 2015, 5:17 AM UTC

Even very advanced mathematics has this feel. You can get a ways with formulae and the like—and it may even be that there is no other language by which you express yourself—but ultimately you build intuition and, as they say, the best proofs arise as a way of making something perfectly obvious apparently so.

There's a strong argument here to be made that all human reasoning is embodied. This isn't the same as a weaker one which might now be left trying to discuss why human's can talk about experiences we can never know—like the behavior at the surface of the sun. Instead, I mean more fundamentally that our brain is one designed to operate in our universe and that our universe plays by many nice rules. Things decompose and move, time flows, causality dominates. It appears increasingly that all of the tools to understand our universe are within these simple forces your brain can't not build an intuition for. To fail to do so would lead to catastrophic inability to function.

Re: Physical Intuition, Not Mathematics (2011)

#33

Post by stuxnet79 » Sun, Jul 05, 2015, 5:29 AM UTC

jrapdx3 wrote: post #11

The problem in discussing this subject is the idea of "intuition" itself because the term doesn't have an unambiguous definition. Meaning varies from one instance to another, even one moment to the next, and disputes arise from imprecise communication. In the example, I think what Feynman is describing, we commonly call "visualization", to be able to "see" the problem in imagination. That is no less a form of abstrac…

> "we never really understand higher mathematics, we just get used to it"

Correct me if I am wrong but I believe this statement is attributed to Von Neumann. It has really stuck with me over the years, just because of how true it is.

I agree that what is important is self-consistency, precision and rigor. Intuition is for the birds. Like you have stated it is ambiguous and often deceptive. Mathematical abstractions enable us to circumvent these pitfalls.

Re: Physical Intuition, Not Mathematics (2011)

#34

Post by jessriedel » Sun, Jul 05, 2015, 5:42 AM UTC

jivardo_nucci wrote: post #24

pdpi wrote: post #13

Earlier quoted context omitted.

QM is counter-intuitive only insofar as your intuition is naturally wired for a classical mechanics world. Given enough practice, QM eventually becomes second nature too. You pretty much _need_ to go through that process to function at the highest levels. There's a good post by Terrence Tao about this topic, I think it was posted here some time ago: https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

Wrong on two levels: Our physical intuitions are Galilean, not classical, mechanics (that is, they are non-Newtonian). For example, our intuitions tell us that an object set in motion eventually slows down and stops. That's Galilean (also termed "folk physics" or "naive physics", usually by cognitive scientists). Most of us had to study formal physics to advance to Newtonian classical mechanics. Quantum mechanics (QM…

There's a lot wrong with what you wrote, but the most glaring is your use of the term "Galilean mechanics".

> Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary.

> the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames.

https://en.wikipedia.org/wiki/Galilean_invariance

Re: Physical Intuition, Not Mathematics (2011)

#35

Post by wyager » Sun, Jul 05, 2015, 5:46 AM UTC

jivardo_nucci wrote: post #24

pdpi wrote: post #13

Earlier quoted context omitted.

QM is counter-intuitive only insofar as your intuition is naturally wired for a classical mechanics world. Given enough practice, QM eventually becomes second nature too. You pretty much _need_ to go through that process to function at the highest levels. There's a good post by Terrence Tao about this topic, I think it was posted here some time ago: https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

Wrong on two levels: Our physical intuitions are Galilean, not classical, mechanics (that is, they are non-Newtonian). For example, our intuitions tell us that an object set in motion eventually slows down and stops. That's Galilean (also termed "folk physics" or "naive physics", usually by cognitive scientists). Most of us had to study formal physics to advance to Newtonian classical mechanics. Quantum mechanics (QM…

I'm not sure why you're being downvoted; you're absolutely right that, intuitively, humans seem to understand physics in a Aristotelian manner. It's quite obvious why; Aristotelian physics is an efficiently computable approximation of real physics that works OK for caveman level technology on earth.

This is very similar to how scientists for a long time believed in classical Newtonian mechanics, because it's a reasonable approximation of the truth at large scales and low velocities.

Re: Physical Intuition, Not Mathematics (2011)

#36

Post by intjk » Sun, Jul 05, 2015, 7:11 AM UTC

reagency wrote: post #28

21p wrote: post #19

Earlier quoted context omitted.

The problem I have always had with the analogy of a person holding up the books with their back is that a table doesn't have to do "work" to hold up the books, but a person does. They table isn't burning any calories.

The table has tension, though. Ad opposed to a table made of paper which would not exert force on the book--the book would overcome the table's tension and set the table in motion until the floor exerted an upward force.

yeah, so think of your muscles as temporarily pretending to act like wood (rigid), and it just so happens that humans have to expend energy in order to carry out this feat.

Re: Physical Intuition, Not Mathematics (2011)

#37

Post by adrianN » Sun, Jul 05, 2015, 7:15 AM UTC

cdoxsey wrote: post #15

This is true of programming too. You're better off learning how to code first and then learning the theory. Once you've got a few simple projects under your belt data structures and algorithms will make a whole lot more sense. You need to learn arithmetic before you learn algebra.

I disagree. Programming languages are just formalisms for writing down the theory and because they're made in part for machines to understand, they are not optimal for humans. I think you're better off learning the theory first using some high level pseudocode before you battle with the quirkyness of real world languages.

Data structures and algorithms are made to solve real problems. You don't have to be able to code up those problems to understand them and see how the algorithm works. Some of the cleverest theoretical CS guys that I know freely admit that they're bad programmers and couldn't implement the algorithms that they describe in their papers.

Re: Physical Intuition, Not Mathematics (2011)

#38

Post by ViViDboarder » Sun, Jul 05, 2015, 7:54 AM UTC

reagency wrote: post #27

abdullahkhalids wrote: post #21

Earlier quoted context omitted.

I agree that there are various patterns of thought that can be labelled intuition. I would like to add on that you have to practice to develop any of these different types intuition. This is I think what Feynman is saying and he is emphasizing the type of intuition (run the experiment in your head) that is very useful for physicists. In every field there is probably a different type of intuition that is useful. In ab…

You don't see the code run in your head, though. You get a feel for how this design pattern works at a glance (intuition), or you carefully step through the code like a debugger.

This was reaffirming. I spend several hours last night unable to sleep trying to solve a problem. As you describe: I had a design and kept throwing scenarios at it and debugged it.

Re: Physical Intuition, Not Mathematics (2011)

#39

Post by batou » Sun, Jul 05, 2015, 8:22 AM UTC

reagency wrote: post #25

stiff wrote: post #22

Feynman also said the following (in the "The Character of Physical Law" lectures): Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics. Newton's statement of the law of gravitation is relatively simple mathematics. It gets more and more abstruse and more and more difficult as we go on. Why? I have not the slightest idea. It is only my purpose here to tell you about this…

This is very important. Many students hit a wall when physics passes beyond lay intuition and whatever may they already know well. Advancing in mathematics is essential to feeling comfortable in advanced physics.

I discovered this last year after arrogantly jumping into the first volume of Feynmans Lectures on Physics.

50 pages in, I decided to take a step back and read a calculus book first, but wait my algebra and trig are crap so back to the basics. So yesterday I hit LCM and GCD applications and factoring which are very basic. So, I'll probably resume the initial book in a couple of years or so...

Re: Physical Intuition, Not Mathematics (2011)

#40

Post by timnic » Sun, Jul 05, 2015, 9:24 AM UTC

batou wrote: post #39

reagency wrote: post #25

Earlier quoted context omitted.

This is very important. Many students hit a wall when physics passes beyond lay intuition and whatever may they already know well. Advancing in mathematics is essential to feeling comfortable in advanced physics.

I discovered this last year after arrogantly jumping into the first volume of Feynmans Lectures on Physics. 50 pages in, I decided to take a step back and read a calculus book first, but wait my algebra and trig are crap so back to the basics. So yesterday I hit LCM and GCD applications and factoring which are very basic. So, I'll probably resume the initial book in a couple of years or so...

Math is wide and deep. You won’t need to cover every topic in math to get going with physics. If you really are interested in physics there are many things in math, which are, well, less important (for doing basic physics). For example LCM, GCD and factoring. I guess, these things are somewhat important in Computer Science, but I never encountered them in a physics problem. So to get started with physics, I would suggest that you focus mainly on analysis (differentiation and integration) and vector algebra. As an addition maybe the basics of complex numbers. This can be learned relatively quickly.

With these you should be able to follow the Feynman lectures or watch the very fine „Theoretical Minimum“ series by Susskind (http://theoreticalminimum.com)