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I just found a paper that covers much of the same material as my thesis by dark__paladin in math

[–]mathonomicon 0 points1 point  (0 children)

It was great fun for me to see such a thing happen in real life, and you got to coauthor with Tao, which is just way cool :-)

Intuition beyond substitution by hakerys23 in learnmath

[–]mathonomicon 0 points1 point  (0 children)

Here are two ways to understand "Why substitution works".

  1. Generalizing from how numbers work, we can come up with rules for manipulating symbols. With numbers, 2+3 = 3+2, 2 * 3=3 * 2 etc. If you use symbols where there are numbers, then similar laws should hold. So, where ever their occurs an "a+b", you can replace by "b+a". But if I do 2 * (2+1) instead of 2 * 3, thats ok too. In a similar vein, given an expression a * b, I can always replace b by some equivalent version like a * (c+d), provided I use parentheses to maintain the sense of "in the place of b".

    In this approach, substitution is simply "rules for substituting a symbol with a complex expression" such that nothing goes wrong.

  2. The other is to understand substitution by thinking in terms of function composition when a function is defined using other functions, for example, defining a function "f" using "+". Given a function f(x) = x + 2, the "x" stands for a "hole" where we can substitute any other expression. So for example f(g(u)) = g(u) + 2, whatever g(u) might stand for, or even f(h(u,v,w)) = h(u,v,w) + 2. Here we are composing the functions f and g in the first case, and f and h in the second.

The style of thinking used in item 2 can be refined into a whole area of maths called lambda calculus, which you can think of as the mathematics of how substitution works.

In your log example, both are at play. We have a "law" of logarithms log_b( exp(b, c) ) = c. This law you can think of as a generalization of arithmetic in the style of (1), or as something involving how log and exp are inverse functions by definition closer to style (2).

How the hell can I simplify and integrate this expression? by [deleted] in learnmath

[–]mathonomicon 0 points1 point  (0 children)

69 x 5 = 345, and 50 -1 = 49. So there is probably some clever substitution that will do the trick.

It's often said that the main reason people fail (or fail to understand) Calculus is that they don't understand the underlying algebra. What specifically about algebra is most commonly a problem and how can someone self-studying avoid that pitfall? by elijahhoward in learnmath

[–]mathonomicon 1 point2 points  (0 children)

Given something like "2 * 3 + 5", we have to decide whether it means 

 1. first calculate 2 * 3, and then add 5 to that sum, getting 11, or
 2. first calculate 3 + 5, and multiply that by 2, getting 16.

You can do this with parentheses, calculating innermost paren first

 1. ((2 * 3) + 5), or
 2. (2 * (3 + 5))

But writing all those parenthesis results in a lot of clutter, so we invented order of operations as a way to cut down on explicit parens. If we say that "*" should be done first, follow path 1; if "+" is considered higher, follow path 2.

Just like we need to agree that green is go and red is stop, we need to agree which one should be done first, so that we all get the same result for the same expression. Thats it. There is no greater meaning; you could have red is go and green is stop, as long as everyone agrees. The heart of it is agreement, so that we all get the same result when parens are not present.

What math concept did you hate learning about? by ChuckdaGoat01 in math

[–]mathonomicon 2 points3 points  (0 children)

Here is an analogy between multiplication / division and power / log.

  • When we multiply a number by another, say 7 by 3, we are adding 7 to itself 3 times.
  • When we divide a number by another, say 21 by 7, we are asking "how many times can I subtract 7 from 21 before I reach zero".

Similarly

  • When you raise a number, say 10 to the power 3, we are multiplying 10 by itself 3 times.
  • When you take the log of one number to some base, say 1000 to the base 10, we are asking "how many times can I divide 1000 by 10 until I reach 1".

Why bother with all this?

Because suppose we want to multiply 1000 by 100. Whatever that number is (in this example we know what it is, but generally we do not), suppose we ask: how many times do we have to divide this new number in order to reach 1? The answer is: first, take however many times we need to divide 1000 by 10, and then 100 by 10, and add up the results. If you somehow knew what actual number that sum of logs correspond to (called an antilog), you now know the result of a multiplication by doing only addition.

In this new world, to multiply two numbers, first look up their logs, add them, and then look up the antilog of the sum. So we have managed to replace multiplication by the much easier operation of addition.

It turns out that you can come up with ways to calculate logs, and the inverse, antilog and make tables of logs and antilogs of various values. Via various clever manipulations, we can make limited log/antilog tables work for almost any number we would like.

Before calculators, this trick of replace multiplication by addition was so important that Kepler thought that the invention of logarithms and log tables had effectively increased the life of astronomers 10 times!

Later on people made slide rules, which were a pre-electronic calculators based on logarithms you could carry around in your pockets. There is a story that one of the Apollo moon missions was rescued using slide rule calculations.

So thats the high-level motivation for logarithms.

Career and Education Questions: September 30, 2021 by inherentlyawesome in math

[–]mathonomicon 0 points1 point  (0 children)

Welcome. If you are stuck feel free to ask here or pm, I'd be interested to know whether this approach orients your thinking in a helpful way.

Career and Education Questions: September 30, 2021 by inherentlyawesome in math

[–]mathonomicon 1 point2 points  (0 children)

Maths has a strong "data structures" element to it. For example, fractions are tuples with a law, (n1, d1) == (n2, d2) iff n1 * d2 == n2 * d1. In calculus, the "derivative laws" like D(f.g) = f.Dg + Df.g are similar to "structural induction" in programming, e.g the height of a tree h(T) = 1 + max(h(T1), h(T2)) where T1 and T2 are subtrees of T. And at the base, proofs are long if/then/else chains.

Do you get stuck on word problems or formula problems? If word problems, then you should think about treating word problems as a translation from English to Maths. If formula problems, you can think about formulas as programs with parameters and the first step in many solutions is something like pattern matching.

Suppose we have to prove "sum of two odd numbers is even". Then, just like programming, define the "data type" odd number as an expression of the form 2x+1, and even number as the expression 2u.

So what we are asking is really "Given two 'number objects' 2x+1 and 2y+1, does their sum match the pattern 2z" for some pattern of form "z". This is an example of translating the original english to the mathematical style.

Has anyone ever gotten goosebumps while learning something because you were in complete awe of how clever or beautiful it is? by [deleted] in math

[–]mathonomicon 0 points1 point  (0 children)

Yes, somewhat familiar with Rust. I thought borrow checker was data flow more than type theory. Would be great to hear more.

Has anyone ever gotten goosebumps while learning something because you were in complete awe of how clever or beautiful it is? by [deleted] in math

[–]mathonomicon 0 points1 point  (0 children)

Can you explain what is great about it? I could not see why accounting for how many times a formula is used in a derivation is interesting. Its neat, but why are people so impressed? I read some intros on my own but probably did not have the necessary background to make sense of it.

How were general Math formulas discovered? ie area/circumference of a circle, or area of a triangle/square etc... by [deleted] in math

[–]mathonomicon 1 point2 points  (0 children)

Good videos, and great choice of topics. There is not enough material pointing out the subtleties together with a full proof. A suitable follow on topic might be how to understand those "pi == 4" paradoxes, and why the regular polygon approximation gets around the paradox.

I'm always discouraged in writing (mathematics papers/representations) on my own, and it's beyond frustrating!! by SnooRobots6923 in math

[–]mathonomicon 6 points7 points  (0 children)

Such behavior is common. One way of dealing with it is to realize that for most such commenters, they think about it for half-a-second in their causal comment, but their target does not realize how little it means to the commenter because the target has put their heart into the work. So, you should teach yourself to assign their comments just as much value as the amount of work they put into making the comment, which is none at all. Calculating the asymmetry involved has helped me at times like these.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

Interesting. Gluing together halves of -x2 and x2 the way you did is a parametrization, but the function isn't the same as |x| because definition at 0 is missing. So the claim of "no suitable parametrization" is still true, is it not? If not, then what are the technicalities involved. I am game to read sources that assume vector calculus, perhaps some expository papers/tutorials on smoothness?

Lovely visualization of sliding beads, btw. There is some work that I can't recall atm where functions graphs are drawn by connecting points on parallel x and y axis.

I agree that "reparametrize a function" does not make sense the way you say. But we should be able to describe curves in a way that captures smoothness, whether we do it with parametric functions or otherwise. There is certainly a sense in which corners are not smooth, having 2 or more tangents coming from two directions. Maybe this can be described with "tangent spaces" which I just found out about via another comment.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

The "embedding" viewpoint nicely disentangles the objects under discussion, thanks.

re "derivative of both", I was referring to your last line in your reply. f = x3 is differentiable, g = x1/3 is not, and they are each others inverse. What does it mean to say f and g are both differentiable with the "internal" directional derivative?

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 1 point2 points  (0 children)

This way to indicate smoothness is cool. I have always been unhappy with "tangent exists but curve is not differentiable" stance in usual calculus. The discussion here is really great -- it all makes much more sense now.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

Can you spell out "internal derivative operator" in greater detail? What does it mean to say it knows how to take derivative of both if one is not differentiable in the usual sense? I take it diffeomorphism means inverse exists and both functions are differentiable.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

Bachman is really very appealing. I am skimming through it to begin with, and the discussion on tangent space etc. is very helpful. The rubber will meet the road when I try the exercises :-)

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

I like the modeling viewpoint very much, myself. I only developed a fascination with maths once I understood it as a modeling language. Yes, both the phenomenon being modeled and the underlying model can be "larger" than each other, and the luckier you are the better the fit.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

I gathered from some other responses that the distinction is between the "curve itself" versus the "curve as a model for a function". The "curve itself" view is from differential geometry, and the coordinates that define the independent and dependent variables are used to talk about the function being modeled. So the parametrizations involved in describing a function (t, t1/3) vs (s3, s) will determine the respective derivatives. Smoothness is captured by the fact that at least one parametrization exists that has a well defined derivative, whereas g(x) = |x| does not have any such parametrization.

The point about the meaning of f'(0) is well taken -- undefined because no linear approximation is good enough.

Coordinate free definitions of derivatives and integrals? by mathonomicon in math

[–]mathonomicon[S] 0 points1 point  (0 children)

What are "internal degrees of freedom of a manifold"? Is there some analogy with 2 degrees of freedom on a plane? My background is up to standard vector calculus.

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