What are you writing about? One of the takeaways for me is that it is good to talk to [someone|anyone] about being stuck. The conversation could lead towards a local minimum, or a local maximum, or even a global maximum.
I haven't read the book, but now want to, and just want to vent something I've been thinking for a long time---this, or something like it, is a really really important project. We desperately need to be able communicate better about calculus.
I sort of agree, but the problem is algebra is fundamentally taught terribly in America, a lot of people get confused on simple operations. Here's the goldstandard essay exploring the failures of math education https://www.maa.org/external_archive/devlin/LockhartsLament....
I am surprised by the usefulness of math concepts as analogies for everyday life. Local vs. global optima, explore vs. exploit, depth-first vs. breadth-first, constraints vs. objectives, necessary vs. sufficient, ...
I also find myself wanting to compose precise statements using "for each", "there exists", "such that", and so on, in a recursive formal logic structure, when discussing non-mathematical topics.
It's hard to tell if those analogies actually help you make real-life decisions, or if they are just a convenient way to state ideas that are also easily stated in non-mathematical language.
I kind of think much of mathematics was constructed to solve problems, so it's not that surprising that it solves problems.
On a deeper level, any optimal internally consistent system will probably be good at reasoning about any real phenomena with the same constraints.
OTOH our mathematics probably reflects human reasoning styles (despite us not understanding but "just getting used to it"), and aliens might have explored different mathematics first (e.g. algebraic geometry might make us the aliens to Euclid).
Meh. Any reasonably broad subject can be analogized to cover any other. I'm more surprised by how versatile baseball analogies are, for instance, considering it's scope is significantly smaller than mathematics.
I like the writing style of this essay. The end of the first paragraph is particularly funny. But to fulfill my contractual obligations for gratuitous criticism as a HNer:
It "uses" calculus only tangentially, and I see simulated annealing and optimization as a computational techniques, not mathematics.
Also but one must be the highest (the “global optimum”) is not neccessarily true; several can be equally highest, e.g. a sine wave.
I agree that saying that he used calculus is a bit of a stretch. However, calculus is essential for all the gradient-based methods (or algorithms) in optimization. Moreover, if you want to analyze these algorithms, you have to know about convex analysis.
I think what the author _meant_ was that he used the intuition derived from calculus to tackle the problem he was framing as a mathematical optimization problem.
Also, you're right, he missed that the global maximum might not be unique in a non-concave function.
I'm a little sad that I never get to use calculus as a software engineer. It's relatively easy to use discrete mathematics, or even something more theory-heavy like type-theory or category theory can be used a bit, but I sadly haven't been able to find much work doing calc, even though I really enjoyed the classes.
Maybe if I started learning to make games that would be an excuse to relearn some physics and I'd be able to use calculus again.
Linear algebra lands in my professional lap surprisingly often, to the point where I ask myself if it's a linear algebra problem fairly early on in the problem solving process. (I'm in GIS)
Calculus is unusual, but it does happen. We do have data with time dependent variables, but usually we don't need anything more complicated than ax + b.
When I took linear algebra and discrete math in college, my linear algebra professor was amazing and my discrete math professor was terrible. I got an A in both classes, but for the life of me I don't grok how discrete math is applicable to programming, but if someone starts describing a problem they have I'll be like, "have you tried linear algebra?" in the "have you tried turning it off and back on again?" voice.
I need to watch one of those lecture series on discrete math.
I recommend playing with TLA+ if you want to see how discrete math can map into compsci; you model everything in terms of discrete states and use logical predicates and set theory instead of "types".
It does seem like the sort of math that mostly gets consigned to writing simulations of real-world phenomena, doesn't it?
I wonder if it has anything to do with how discrete software is. A program executes in lurches, not as a continuous process. You sort of end up dealing with Riemann rectangles instead of curves, when you look at the time domain.
That resemblance is probably just a coincidence, though; maybe it has more to do with the fact that most programs react to input after the fact instead of trying to model the immediate future and act based on predictions.
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[ 21.5 ms ] story [ 908 ms ] threadSketch a breadth-1st near-random (high heat/entropic) exploration map/terrain of conceptual/creation structure optima.
Over time, let the system cool.
Slowly raise the threshold of improvement-gradient acceptability (increasing negative value). This confines your exploration.
Ultimately end in a cooled state, going only directions of direct improvement.
I also find myself wanting to compose precise statements using "for each", "there exists", "such that", and so on, in a recursive formal logic structure, when discussing non-mathematical topics.
It's hard to tell if those analogies actually help you make real-life decisions, or if they are just a convenient way to state ideas that are also easily stated in non-mathematical language.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Wigner, http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html wikipedia article https://wikipedia.org/wiki/The_Unreasonable_Effectiveness_of...
I kind of think much of mathematics was constructed to solve problems, so it's not that surprising that it solves problems.
On a deeper level, any optimal internally consistent system will probably be good at reasoning about any real phenomena with the same constraints.
OTOH our mathematics probably reflects human reasoning styles (despite us not understanding but "just getting used to it"), and aliens might have explored different mathematics first (e.g. algebraic geometry might make us the aliens to Euclid).
It "uses" calculus only tangentially, and I see simulated annealing and optimization as a computational techniques, not mathematics.
Also but one must be the highest (the “global optimum”) is not neccessarily true; several can be equally highest, e.g. a sine wave.
I think what the author _meant_ was that he used the intuition derived from calculus to tackle the problem he was framing as a mathematical optimization problem.
Also, you're right, he missed that the global maximum might not be unique in a non-concave function.
Maybe if I started learning to make games that would be an excuse to relearn some physics and I'd be able to use calculus again.
Calculus is unusual, but it does happen. We do have data with time dependent variables, but usually we don't need anything more complicated than ax + b.
When I took linear algebra and discrete math in college, my linear algebra professor was amazing and my discrete math professor was terrible. I got an A in both classes, but for the life of me I don't grok how discrete math is applicable to programming, but if someone starts describing a problem they have I'll be like, "have you tried linear algebra?" in the "have you tried turning it off and back on again?" voice.
I need to watch one of those lecture series on discrete math.
I wonder if it has anything to do with how discrete software is. A program executes in lurches, not as a continuous process. You sort of end up dealing with Riemann rectangles instead of curves, when you look at the time domain.
That resemblance is probably just a coincidence, though; maybe it has more to do with the fact that most programs react to input after the fact instead of trying to model the immediate future and act based on predictions.