Clinamenic

θερμοδυναμικός καπιταλιστής
Feynman brought something similar up in an interview, how sometimes he had moments of extraordinary insight, and he became compelled to understand the conditions of these extraordinary moments, in order to more readily replicate them.

I'll keep trying, keep pulling it out of its ineffability.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
Such an ability would be a sort of intellectual fertilizer, a means of reconciling generalism with expertise.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
If you want to get into physics, check out Feynman. He contagiously wore the joy of physics.
 

wektor

Well-known member
Feynman brought something similar up in an interview, how sometimes he had moments of extraordinary insight, and he became compelled to understand the conditions of these extraordinary moments, in order to more readily replicate them.

I'll keep trying, keep pulling it out of its ineffability.
I think it is definitely the case with mathematics, you slowly gain knowledge and build up some chunks of skill when it comes practically to solving problems, at some point things click and everything you learned until that point intuitively makes sense as a whole.

I think these would be the three critical parts: knowledge, skill, intuition.
3Blue1Brown is great for building up intuition (and that's what he says very often).

As for me I feel like I always very much overlooked the overlap between knowledge/skill, which is very much apparent when I encounter a book in maths, actually a common way to study the discipline. I can read through, but can I read with comprehension? Not yet, I'm afraid.

Due to some stuff I'm reading right now (and my recent obsession with C C Hennix) I can relate to OP a little bit, trying to slowly catch up on category theory and topology recently, though I'm lucky to have a very kind friend to help me out and recommend good sources.
I think in the scope of this thread external support might prove fruitful, as searching for materials on a discipline you're not exactly familiar yourself does not sound optimal.

Perhaps we could recommend some readings to @Clinamenic ? What's your field of interest? Where do you feel you lack the most?
 

IdleRich

IdleRich
On the cusp of a significant conceptual breakthrough in my understanding of physics. Was perusing Physics Forums and started on a wikipedia chain leading to "Jerk" which is the change of acceleration in respect to time, or m/s^3 (acceleration itself (m/s^2) being change of velocity in respect to time, and velocity (m/s) being the change of position in respect to time).

I have been struggling with trying to understand acceleration as m/s^2, in the most general way I possible can. I have been more easily able to understand velocity as m/s, but when the denominator gets exponentiated my intuition drops off accordingly.

Just now, between the wikipedia pages and the wording employed by some members at Physics Forums, I managed to get a better comprehension of the concepts at hand. If acceleration is the change of velocity over time, it can be considered as (m/s) x (1/s), which comes out to the same thing as m/s^2.

(edit: for the record, the following wording is my own)

The change of the change of position in respect to time, in respect to time.

(m/s) x
(1/s)
This kind of unit analysis can be quite useful in mechanics - in lots of physics in fact.
This kind of differentiation is considered to have been invented/discovered (which?) by Newton and Leibnitz independently of each other at around the same time. Quite interesting I think. Hence two notations.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I think it is definitely the case with mathematics, you slowly gain knowledge and build up some chunks of skill when it comes practically to solving problems, at some point things click and everything you learned until that point intuitively makes sense as a whole.

I think these would be the three critical parts: knowledge, skill, intuition.
3Blue1Brown is great for building up intuition (and that's what he says very often).

As for me I feel like I always very much overlooked the overlap between knowledge/skill, which is very much apparent when I encounter a book in maths, actually a common way to study the discipline. I can read through, but can I read with comprehension? Not yet, I'm afraid.

Due to some stuff I'm reading right now (and my recent obsession with C C Hennix) I can relate to OP a little bit, trying to slowly catch up on category theory and topology recently, though I'm lucky to have a very kind friend to help me out and recommend good sources.
I think in the scope of this thread external support might prove fruitful, as searching for materials on a discipline you're not exactly familiar yourself does not sound optimal.

Perhaps we could recommend some readings to @Clinamenic ? What's your field of interest? Where do you feel you lack the most?
Wonderful. Thank you.

3Blue1Brown has been important for me so far, namely his series on linear algebra, which I will probably rewatch several more times.

One topic I'm struggling with is how mass relates to energy. Also how gravity relates to mass.

Perhaps it is a common feeling to feel that my understanding of these things is somehow scale dependent and eventually "breaks" as we inspect finer and finer phenomena.

Another area is Dirac notation, and complex numbers in general. I know the a + bi formula, pretty sure thats it, and I can see that for every real number a there is a whole dimension of complex numbers a + bi, but I don't understand why this is useful. A breakthrough in understanding quantum mechanics seems dependent on such a breakthrough in these maths.

Also have difficulty with quantum numbers, the Higgs boson, etc. Not sure how I should be conceptualizing these things, again largely because I;m not sure how to conceptualize mass and energy and gravity.

Also have difficulty conceptualizing EM waves, unless they can be considered as their own thing as the standard model, to me, implies.
 

IdleRich

IdleRich
I think that complex numbers were discovered/invented, at least in part, due to trying to find a formula for the general solution of cubic equations. You all did the general solution for a quadratic at school but the cubic one is a lot harder and took people centuries to crack (even though all cubics do obviously have at least one real solution, unlike quadratics). But one guy working on it kept finding it coming to an impasse where he had to square root a negative - which obviously couldn't be done with the numbers we had then. But he thought "what if I just write square root of minus one and move it around and manipulate it like normal numbers and hope that it gets squared later and then I can say I've just gone back to -1 again" and I guess he did that and came up with a formula... and it worked!
And then people used that trick for further manipulations of square roots of negatives etc
 

Clinamenic

θερμοδυναμικός καπιταλιστής
Huh, so a departure from real numbers that is permitted on the condition of a return? Fascinating.

Don't know anything about quadratics and cubics, so theres something else.
 

IdleRich

IdleRich
Huh, so a departure from real numbers that is permitted on the condition of a return? Fascinating.
I wouldn't say that that IS the rule, but I think that at the time, as they had no concept of complex numbers, that was kinda the way he was thinking.
 

IdleRich

IdleRich
I suppose that at the time if someone had said "solve this cubic for me" and he used complex numbers to come back and say "The answer is 5, 7 and 21" then that would have been one thing.... but if he came back with "the answer is the square root of minus three" then no-one would have known what that meant.
 
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IdleRich

IdleRich
Complex numbers are used for all kinds of things since then though. You can also define the trigonometric functions in terms of complex numbers.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
So they are used in cases, which isn;t to say only in cases, where you need to work around otherwise uncomputable things like square roots of negatives?
 

IdleRich

IdleRich
Like Is say, you can write trig functions in terms of complex powers (if you can get your head around that) of euler's constant - so complex numbers arise in anything that has trig in it.
Also you get Euler's Formula
e^{i\pi }+1=0
 

Clinamenic

θερμοδυναμικός καπιταλιστής
Can i be considered a tool that isn't a number, but in a weird way an operator?
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I'm starting to see it as a sort of portal out of and back into computability.
 

IdleRich

IdleRich
Ah, but thats still enough for me to start getting a better grasp of them. Thank you.
No prob. I think in general a lot of maths was invented when they faced a problem and the tools they had didn't solve it. So they created something new to deal with it... and then often took that thing and used it in other contexts.
Complex numbers were invented by someone trying to solve cubics but does that mean that they exist less than the numbers we had before that?
I think we had counting numbers, then maybe they had to split things between people so they invented fractions, then probably they had to deal with debt so they invented negative numbers, then the Pythagoreans pointed out that root 2 was none of the above so they came up with irrational numbers, then further on, as discussed, they created complex numbers. But do any of them really exist or not I dunno?
 

IdleRich

IdleRich
Can i be considered a tool that isn't a number, but in a weird way an operator?
I dunno.
I'm starting to see it as a sort of portal out of and back into computability.
Don't get too hung up on this I think. They were invented to get past an impasse but as soon as they were out of the box then they took on a life of their own. They are as computable as anything else now, probably best seen as one of a list of types of numbers that were called into existence as they were needed (see post above).
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I don't think math has any significance beyond our use of it to understand the universe. I don't think of it as a universal source code that we discovered.
 
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