
Physical Intuition, Not Mathematics (2011) - k0doque
http://realphysics.blogspot.com/2011/06/physical-intuition-not-mathematics.html
======
stiff
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 fact. The burden of the lecture is just to
emphasize the fact that it is impossible to explain honestly the beauties of
the laws of nature in a way that people can feel, without their having some
deep understanding of mathematics._

~~~
reagency
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.

~~~
batou
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...

~~~
timnic
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](http://theoreticalminimum.com))

~~~
batou
Thanks for the pointers. Much appreciated.

I'm doing a full review of mathematics at the moment. Not in depth, more of a
"here's an application of the GCD function" so I know what tools to use to
solve specific problems. All this is beneficial for the day job as well who
expect to see some value from my time spent even though I'm not being totally
honest with the objective to them. Realistically I want to think abstractly in
the terms of mathematics and develop some intuition.

Was completely unaware of the Theoretical Minimum series. Thanks for that.

Edit: I'm reading _Mathematics: From the birth of numbers_ by Jan Gullberg as
a text. Wonderful book. Covers just about everything and is beautifully
written by a non mathematician with no assumptions spared and no education
target. In fact the forward is mainly bitching about the education system.
Slightly worried I will get distracted by this book but that's never a loss!

~~~
timnic
I do not know "Mathematics: From the birth of numbers" but judging from the
Amazon quick view it seems to cover a lot of ground (BTW: one thing I missed
in my list are the basics of differential equations).

Over 1000 pages is quite a long read, though. I never managed to read a
(science) book as big as that from cover to cover myself. One thing I learned
through the years is to never use only _one_ book for learning. Books have
different styles and not every style fits to every student. Additionally one
book might be good at one specific topic and weak on another. So nowadays I
always use a couple of books (or online resources) to learn a new topic.

~~~
batou
Its huge yes but a lot of it is fluff and history. It does serve to keep it
interesting however.

Quick page shot to show the scope and density:
[http://i.imgur.com/sV1WYFd.jpg](http://i.imgur.com/sV1WYFd.jpg)

I have a number of other books as well that I use as a reference as well so no
problems there (calculus for the practical man has some different insights).
Oh and betterexplained.com.

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jrapdx3
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 abstraction that is often vital to problem-solving. Of course,
not every problem yields to this approach but it is a powerful feature of our
basic cognitive tool set.

Einstein wrote that his early success in formulating his idea of special
relativity was the outcome of his intuition about the physical properties of
light, etc. he studied. Later on, the mathematical abstractions became more
powerful, and at that point intuition about the "physical" nature of phenomena
was insufficient for understanding.

But I think there are forms of intuition that apply to very abstract ideas, or
what seem to be so to us. I once heard a physicist say "we never really
understand higher mathematics, we just get used to it". Feynman would probably
have agreed with that sentiment.

"Getting used to it" is really the equivalent of developing an intuition about
the subject. I remember first learning about programming recursive functions,
mind boggling in the beginning. After a while, it began to "sink in", that is,
it became intuitive, I no longer had trouble "seeing" how it worked. The key
is familiarity, something once strange is now digestible.

So there's nothing binary about intuition, it covers many forms of thought,
and incorporates reasoning about emotion, having a "feel" for the problem in
question. There are limits to our abilities, at the highest level it's genius,
but there are no clear boundaries.

~~~
abdullahkhalids
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 abstract math maybe 'getting used to it' very fast is very useful. In
geometric math maybe being able to visualize objects in space is useful. In
programming maybe seeing the run of the code in your head is useful.

~~~
reagency
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.

~~~
ViViDboarder
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.

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cgag
I just started reading this book, "Thinking Physics", which teaches physics
the way Feynman talks about, by trying to build intuition. It asks short
questions like Feynman's about the table, and expects you to think about the
answer for a while, before giving it to you and presenting the physics behind
it.

[http://www.amazon.com/Thinking-Physics-Understandable-
Practi...](http://www.amazon.com/Thinking-Physics-Understandable-Practical-
Reality/dp/0935218084)

------
tel
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.

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krastanov
While this is a lovely sentiment and a perfectly fine way to approach the
sciences (and also engineering oriented disciplines), not all physicist think
that way.

There is a lot of beauty in deriving laws of nature without relying on
physical intuition. A lot of beautiful results are based on purely requiring
laws to be self-consistent and seeing that only one possible law is self-
consistent. For instance check out what Scott Aaronson says about probability
in quantum mechanics. While Feynman in his famous lectures just says that
quantum mechanics is counter intuitive and you are not supposed to truly
understand it, Scott Aaronson uses math to explain how to correct your
intuition (and I am stressing, this is not just about learning the math, it is
about basing your intuition on the math, not on the everyday experience).

~~~
pdpi
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...](https://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

~~~
jivardo_nucci
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) is completely non-intuitive, at least as far as
intuition about either folk physics or Newtonian classical physics is
concerned. IIRC Feynmann says as much in his book "QED: The Strange Theory of
Light and Matter", Chapter 28 beginning:

"I think I can safely say that nobody understands quantum mechanics. —Richard
Feynman

The quantum theory is not explicable in commonsense terms..."

Of course Feynman used his real-world physical intuition all the way through
to his most abstract work. In one case he characterised the internal structure
of the proton as being like "marbles inside a tin can." Try and write those
equations!

~~~
wyager
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.

~~~
jessriedel
His comment has technical errors, but more importantly it just doesn't address
the parent's argument. It's a non sequitur.

------
edtechdev
Related to this, there's a fair amount of research on the connection between
embodied cognition and the learning of physics (and mathematics), as well as
examples of 'embodying' physical forces and laws.

An example from Hans Freudenthal is from a standard physics question. If there
are books on top of a table, what are the forces acting on the books? Most all
students draw the downward force of gravity, but some forget about the force
the table exerts upward back on the books. You can have students get on their
hands and knees and put books on their back, or have them lie on their back
and hold up books with their hands and arms. They 'embody' the table, in a
sense. When you add a second book, you feel that you have to exert more effort
(which correlates with force) to hold the books up.

~~~
21p
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.

~~~
reagency
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.

~~~
intjk
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.

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UhUhUhUh
A problem is that the reverse is also true. Simple example: stretch a rope
around the earth that fits tightly and add 1 m to the length. Can a cat go
under the rope (it's always a cat). Intuitively that seems like a "No". The
simplest of formulae says "Yes" ( 1m/2pis = +/\- 15 cm). Here, pretty much
everyone has to resist intuition and trust numbers. I think the best way to
conceptualize intuition is some sort of unconscious sub-process carried out by
the brain at all times that has to be shut down or de-prioritized most of the
time to allow for hard, "empirically-validated", processes to dominate the
global activity (similar to early life "pruning"). As all neuronal processes,
these need to be activated to remain available.

~~~
harperlee
Intuitively it can also be answered: suppose you pinch the rope, so it is
tight (you step on it and pull from between your two feet, for example). You'd
have a handle for the earth that will go to about just above your knee. A cat
fits!

~~~
UhUhUhUh
Good point. That's the power of thought-experiments! It's about stimulating
intuition.

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crimsonalucard
Does anyone have a good theory for what they think intuition is?

I'm interested in process. What is the mapping algorithm that transforms
problem into solution? While intuition appears to be magic, I believe that
there is a very concrete process happening in our subconscious.

My personal guess is that we're transforming the problem into a format more
suited for different modules of our brain to process.

For the table problem we apply 2 transformations. One is referred to as a
calculation, the other as intuition.

"Calculation" involves transforming the table into symbols (mathematics) for
the language/logical part of our brain to process; the other transformation
involves turning the table into a sort of fuzzy 3D visualization for the
imaginative/spatial part of our brain to analyze.

Both transformations yield transformed results. Symbols yield a
symbolic/numeric solution, a fuzzy 3D visualization yields an equally fuzzy 3D
solution.

Funny how two different process that are both seemingly systematic are called
different things. Why is one called intuition and the other not?

~~~
asgard1024
I am no expert, but it seems to me that intuition is what the machine learning
algorithm develops as a model.

The model itself cannot explain how it arrived to its conclusions; that could
be done with the training data which are long gone.

Similarly to machine learning, humans can develop intuition about things by
learning and training in the subject (that's why I believe rote learning is
actually quite useful).

Just like with intuition, it crucially depends on (and varies with) the input
data (experience) and there can be different models, but successful models
(those that give good results on training data) are quite similar in
appearance.

Of course, the big disadvantage of intuition is that you cannot explain it to
others, even if it works. They have to believe that you are expert and made
correct judgements (that you have correct model). That's why science (and
especially mathematics) has tried to formalize the process, so that people
could double check the reasoning and wouldn't have to rely on expert
authority. That's why the two processes are called differently, I think.

~~~
crimsonalucard
I agree. This is a good theory.

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brak2718
The technique served me very well doing physics and engineering in an earlier
life. I struggle now with the chaos of software systems because I'm unable to
"feel" what's going on.

~~~
dack
It seems like a common intuition in software is the "code smell", where you
can just look at some code and tell if it will be "maintainable" or "good",
for some value of that.

I've come to think of it more as a "familiarity" though - where if you're
familiar with writing stable and maintainable systems, you'll get the right
smells - but some people have opposite feelings about systems, and it seems
like they are people that have the opposite type of experience as well.

Other types of intuition I think are based mostly on pattern recognition. If
you have a system that makes a set of choices that follows a well known
pattern to you, your brain can start predicting what has been done where, and
with enough confirmation, can start feeling confident about the behavior of
other parts not yet scrutinized. Once again, if things are done in an
unfamiliar way, all that evaporates and one needs to fall back to looking at
the low level strategies and techniques.

------
cdoxsey
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.

~~~
adrianN
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.

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ma5sudaca
I think this is interesting because the implications go beyond just physics
and mathematics.

Sometimes we learn things in one domain that could be easily applicable in
others, yet since we've never practiced said things in another context, it's
hard for us to make this connection. Often, all we need is a little nudge in
the right direction, which could open to us a new world of insights that we
didn't have before.

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marincounty
I had a bad anxiety attack one time. The room I was in felt like it was
undulating. Everything started to become fluid like. I can't explain it, and
should probally draw it, but years later I still wonder if that's the way the
real world actually looks?

Never told anyone--out of fear it could happen again.

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dschiptsov
Intuitions are insights (hints) from so-called ancient, non-verbal (pre-
linguistic) instinctive "knowledge" or "genetic memory". It is not only the
kind of knowledge of how birds "know" how to make nests or men know to run out
of building when earthquake happen (without any prior training), but also
intuitive knowledge about the nature of reality, properties of physical
environment, which has been "trained" before any language, even before human
species has been evolved (we share "pre-cortex" brain centers with our
ancestors which were shaped by environment). This is where intuitions are
coming from.

~~~
reagency
That is "instinct", a different concept from "intuition", thought overlapping
in some aspect.

------
wfo
It strikes me as kind of silly the way he tries to separate the mathematics
(i.e. what he calls simple calculations) from the physical intuition; all
mathematics beyond say sophomore year in college is very intuitive but at an
abstract level (and the earlier stuff too, they just teach the early
calc/stats classes to scientists/engineers so they drop the intuition and make
it about pure mindless calculation), mathematicians are primarily intuitive
creatures who have just developed intuition over years of hard work about
abstract objects which are far more difficult to intuit about than, say, a
table.

------
newsposter123
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.

~~~
gh02t
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 discuss something, discovering its properties becomes
much more straightforward.

~~~
tjradcliffe
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).

------
fiatjaf
Isn't this obvious? Why does someone have to say it?

~~~
crimsonalucard
The table is obvious because it's used as an example. Developing intuition for
something list say... relativity is not as obvious.

~~~
fiatjaf
Relativity is impossible to imagine and intuit, also particles and quantum
phaenomena. Physicists always tell the public to "not try to imagine" these
things, because that's the natural and obvious thing to do: to imagine and try
to intuit.

This student at the example is probably crazy or fake.

