This statement is misguided. A lot of physical phenomena doesn't lend itself to intuitive understanding. Quantum Mechanics is a great example of this. You cannot expect subatomic particles to behave the same way that macroscopic objects do (and macroscopic behavior is what humans find intuitive because we experience it on a day to day basis).
Its the same misguided thinking that leads to the popular PopSci explanation of electron behavior being both wave and particle like. Why should we try to categorize the behavior a subatomic particle (an entity so distant from our experiential reality) as being analogous to one of two macroscopic entities?
I totally get the desire for intuitive understanding, and it should be encouraged, but sometimes you just have to put intuition aside and come to conclusions with pure mathematical reasoning.
Isn't this just a misunderstanding of what we mean by intuitive? It's possible to have an intuitive understanding of a thing that's not "intuitive" to humans.
Exactly. Just imagine the quantum world as a bunch of liquids floating around carrying their associated particles, and you have an intuitive grasp of (the debroglie bohm interpretation of) quantum physics.
I'd even be more ok with that, if mathematicians used more rigor in their notation. Math developed in a time where it was tedious to write this out by hand and a lot was hidden in abbreviations and assumptions. One thing that programming gets very right is that it is way more explicit (although obviously not perfectly) the behavior behind any notation.
It's one thing that Structure and Interpretation of Classical Mechanics (which I haven't read) got really right in concept. It's a shame, that approach hasn't been adopted more widely.
Take for example page 7 of the pdf with the heat equation. Even the description of the Laplacian is wrong. The value isn't the average of the points surrounding it. But if this simple function were written in python as a "update" function over a multi-dim array of temperature values it'd be clear exactly what's happening.
Another example of simple inconsistency in notation: superscript, does that mean squared or not? When writing two terms adjacent is it multiplication or an operator being applied? Where is the behavior of the operator defined that a student and "read the code"?
English is an extremely verbose and imprecise language. Mathematicians replaced a verbose imprecise language with a terse imprecise language. It's about time for rigorous fields to take the last step and introduce precise concepts using a precisely defined language.
> The second part describes the long-run behavior of the differential equation
> The temperature value that this point takes is the average temperature of the points surrounding it
And this is not correct. Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren't describing and evolution if its inaccurate for all points except a final state.
On top of that, even during the evolution the temperature isn't what's taking on the "average" of surrounding points, it the change in temperature wrt time that's being change by the average of points. And yet again, it's not an average of the surrounding temperatures is and average of the surrounding differences in temperature.
And this interaction we've had highlights exactly my point. English is an impressive language, and Mathematics is full of hand wavy explanations that come with simplicity context that isn't explicity and precisely defined. Mathematicians are used to it, and of course it's learnable like anything. My point is it's not necessary, it evolved in a different time with different constraints. Its a similar example of "the medium is the message" - math evolved when the only way to write was by hand, and duplication of definitions was manually expensive. We don't need these handwavy shortcuts now - we can make it easier for students to learn by being precise and providing definitions for them to see exactly what's happening - not expect them to learn from trying to recall every imagined context. And the fall back of "I had to learn it this way, so they should too", is a horrible excuse.
> Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren't describing and evolution if its inaccurate for all points except a final state.
The average of the surrounding points is the attractor temperature for the system. It is an asymptote which the temperature of the point is moving towards. It's like saying an oscillator (such as a spring) wants to be at neutral, even though it never comes to rest at neutral.
I'm not engaging with your larger point, I'm just quibbling with you saying page 7 is wrong. I think that "takes" in the english description says that the temperature approaches the value over time, whereas you interpreted "takes" as referring to the temperature at every point in time.
> I'm not engaging with your larger point, I'm just quibbling with you saying page 7 is wrong. I think that "takes" in the english description says that the temperature approaches the value over time, whereas you interpreted "takes" as referring to the temperature at every point in time.
I hear you, and I think we've probably approaching the end of the productive part of our conversation.
I do want to mention that the quote above, and interpretation of "takes" is exactly my larger point though. These definitions are all sloppy and prone to interpretation. Precise definitions would eliminate the need for all of this.
And since I can't help myself, thinking about this a little more, even your interpretation above is either faulty, or has to change the definition of "surrounding". If I have a point of average temperature, a doughnut or sphere of warmer points a small distance immediately around it and then the majority of all other points around that being colder, then the asymptote is actually towards the average of the of outer colder points, not the surrounding warmer points.
QM can be understood intuitively, the problem is we are still teaching the bullshit 'wave particle duality' when we basically know that fields are what is real.
There's also no such thing as 'observation'. Fields interact by a precise and well-tested mathematical function. That function does have the effect of mutating the state, but why should there be such a thing as pure observation? There's nothing magic about conscious observation.
Perhaps the comment was intended to convey the very different potential for intuitive "understanding" between visualisable classical theories and non-visualisable quantum theory.
At a fundamental conceptual level it does not map to anything in our normal experience - attempting to force analogies etc can end up confusing things further.
I agree with your original observation btw, but it's important to recognise the limits we may be bound by in explanation and comparison.
