[eellpp]

Last year, i saw the videos by 3Blue1Brown and inspired by it went on to read some of the standard text books like linear algebra application by Strang. I had seen the Strang videos earlier but somehow did not follow it through. This time however my perspective was changed. I was approaching the subject through the lens of intuition and simplicity. Wherever i found something challenging, i waited for the next day to again re-do it (because i know that this should not be that complex to understand or the understanding is wrong). And to my surprise, again and again, the difficulty was in my rigidness in understanding. The next day , or even later during the day, when my mind was fresh again, i can reason through the concept and get the intuition behind it.

Since then i have seen the Strang videos again and again. Beginning to end. Read the book chapter by chapter and exercises by exercise. And what a delight it had been. And then i jumped upon Joe Blitzstein's probability lectures. What a blast ! Is there a list of teachers like these there, who in the pretext of teaching algebra/probability etc are in reality wiring up our thinking process in ways immaterial to subject they are teaching. Many of us don't want the material to be too casual/layman terms (which hampers self understanding as its no challenging anything within us) and not too rigid (where we cannot break through the challenge).

[espeed]

This time however my perspective was changed. I was approaching the subject through the lens of intuition and simplicity...And to my surprise, again and again, the difficulty was in my rigidness in understanding.

Thank you for penning those words. I hope people realize the significance of what you just said.

Perspective is key. You could say it is the KEY -- the key insight into unlocking everything else. I had a similar experience in 2009, and once you have the epiphany -- once you experience the awe of a shift and recognize the implications -- it's like your mind becomes unshackled. You realize you have been blind, and you've just learned to see. And in that flash you gain a deep, visceral understanding of what Alan Kay means when he says, "A change in perspective is worth 80 IQ points," and "We can't learn to see until we admit we are blind."

"Life's Illuminating Perspective" (2009) http://jamesthornton.com/manifesto

[j2kun]

> the difficulty was in my rigidness in understanding

At what point did you realize this? Like, could you provide a specific example of a topic you thought was hard at first but later came back to and realized was all about the intuition?

[Retra]

Not the OP, but...

I was just reading Landau & Lifshitz' "Statistical Physics", and can reflect on a series of thoughts that may elaborate on how intuition plays a role in the enjoyment and understanding of complex material. I've been meaning to write it down anyway...

On page 3, the book says "A fundamental feature of this [closed system/open subsystem] approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state." When I first read this, I thought "non sequitur, but whatever, I'll continue..." Now, the context of this quote is that the authors are trying to explain why statistical methods work at all. And they said prior that we start with laws that apply to 'microscopic' particles and use statistics to generalize to 'macroscopic' systems.

However, the second time I read this, I kept thinking it must be backwards. We didn't understand the motion of (classical) protons before understanding the motion of macroscopic balls. So we had to have been operating under the assumption that the macroscopic laws must apply to microscopic objects, and then require that the must also be reproducible macroscopically through statistical methods. That is, we require that these laws be invariant across the microscopic-to-macroscopic transition. But to do that, we have to use a framework which expresses such a transition. So, for instance, if we are reasoning about the motion of a ball, we have to translate our laws into laws over the motion of some statistical model of the ball. Say, it's center of mass. And with this concept in hand, we could write laws that apply to both the macro and micro worlds, since a 'center of mass' is a macro-micro-scale-invariant abstraction. So we partition the space of all possible laws of nature, and chose to work only in that partition which encodes things we can actually know about nature -- the partition identified by the macro-micro-scale-invariant.

So re-reading that passage, it is now not a non-sequitur for me. Now it says "because the interactions with the outside world are so complex, we could not hope to predict their influence. Thus we are justified in using random variables to model their influence, and concepts that derive from the use of random variables to ensure we have complexity-scale invariance when we formulate our laws of physics." And this is not a non-sequitur to me. It follows directly from the meaning of the word "random." Of course statistical methods work when the complexity of a system is indistinguishable from randomness.

--

And this whole line of thought generalizes (albeit informally.) For instance, the meaning of words is an invariant across a long thread of contextual translations, and these invariants are used the same way: to partition the space of all possible meanings in such a way that one partition contains all of the 'knowledge' imbued by that word, and thus you can navigate a narrower space of meaning to find the intended and/or correct one. Gives me a certain brand of appreciation for good poetry.

Or -- my girlfriend -- who recently told me that she loved algebra but couldn't understand trigonometry. I tried telling her that the algebraic transformations were invariant-preserving operations selected because they conformed to known laws about 'functions' like addition and multiplication which have commutativity and identity laws, and that trigonometry was no different: different functions, but all of the algebraic transformations you needed were selected from the laws of trigonometry with the purpose of maintaining the exact same invariants. (Not that she cared much, to be honest...)

And on and on... I could probably talk for days about all the different ways every subject can be reduced to transformations and invariants and how they are used to solve problems.

[noobiemcfoob]

> I could probably talk for days about all the different ways every subject can be reduced to transformations and invariants and how they are used to solve problems.

I find myself partial to this type of world view too. I believe it is part of the appeal of functional programming, at the basest level, to shape the programming model into transformations (functions) and invariants (state).

[j7ake]

Insightful comment thanks for your anecdote.

[j2kun]

Thanks for writing that.

[edge17]

Could you link to the probability lectures you're referring to?

[espeed]

Presumably...

Statistics 110: Probability - Joe Blitzstein, Harvard University http://projects.iq.harvard.edu/stat110/youtube

[eellpp]

Thanks. Yes, that's the link. Joe is one of those teachers, who try to simplify the concepts while preserving the elegance of what is described. Kind of like minimalistic simplicity in teaching. While teaching, every time he would emphasise the intuition behind them. He avoids the complex formula's/algebra, calling them ugly ... and often presents one line proofs to them !!. After getting through the first few of lectures, you seem to get the underlying trend in his approach. That all the complexity is just hand waving over the the simplicity of underlying concepts. (Sometimes i think if the maths is complex to explain a phenomenon then we are not using the correct theory to explain it. But i am not a mathematician .... )