My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy.
An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?
Oh and if you really love that dumb, sophomoric ending, you can also get Return to Monkey Island, which is yet another game Gilbert ruined with this tired trope.
And that's why Curse of Monkey Island is and will always remain the best MI game.
> Oh and if you really love that dumb, sophomoric ending, you can also get Return to Monkey Island, which is yet another game Gilbert ruined with this tired trope.
Isn't it a bit different with Return to Monkey Island? The story is told to kids by Guybrush and Guybrush being an unreliable narrator, it's uncertain how much of the story was real or an imagination by him.
The last scene though shows that something real somewhat similar to what Guybrush told did happen.
Also it completely retcons the ending of Monkey Island 2.
Can't say i disliked Return to Monkey Island much. I liked the art style and the characters. The story wasn't amazing.
MI1 will always be the top for me i guess, but i still remember almost every scene after all these years, so it's not really fun to play again and again. You can play it on almost anything by now [1]
Having working in that area a little bit, simple heuristics to know if someone has dumb ideas about advertising are whether they: a) think it's some kind of magical MK-Ultra-level mind control technology, b) think the business that pays them for a living could exist without any form of advertising. Once that's acknowledged as fantasy you generally get into goal-post moving territory about what is or isn't really advertising but that's purely academic.
Lagrangian / Hamiltonian mechanics, the principle of least action, always seemed neat, in L&L and other places I encountered it, until I tried doing exactly what you're saying: gaining an intuitive understanding. At that point it just never made sense to me and seemed like a gratuitous deus ex machina that happens to work beautifully but for no apparent reason. You won't be surprised to learn I dropped out of my STEM program shortly after, though I keep a keen interest in the topic.
About the stationary action concept:
Yeah, it looks impenetrable, but here's the thing: there is a way of looking at it from just the right angle, and then becomes transparent.
Part of the story is this: the actual criterion is: the true trajectory corresponds to a point in variation space where the derivative of the action (derivative wrt applied variation) is zero.
In the cases examined when the concept was first introduced I suppose that in those cases the derivative-is-zero point was seen to be a minimum. From there, I suppose, came a supposition that there was some form of minimization at play.
However, within the scope of classical mechanics there are also classes of cases such that at the point in variation space corresponding to the true trajectory the action is at a maximum.
The above, and other aspects, are discussed in a resource that I created.
In the resource the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond.
About interpretation:
As we know: motion along the true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. As we know: that property is known as the work-energy theorem.
The criterion derivative-wrt-variation-is-zero corresponds mathematically to the property: rate-of-change-of-kinetic-energy-matches-the-rate-of-change-of-potential-energy.
In the resource a two stage process is presented:
- Derivation of the work-energy theorem from F=ma
- Transformation from the work-energy theorem to classical mechanics stationary action
Of course: when you look at the work-energy theorem you wouldn't expect that it can be transformed to classical mechanics stationary action.
The transformation consists of multiple steps. In the resource I present it step by step; for each step the logic and consistency is readily recognizable.
For me, having the breakdown into mathematical elements available changed my whole perspective on classical mechanics stationary action.
I hope I can persuade you to check out the resource
More than twenty years ago, I quit a program that taught math/cs/physics (the notorious French "classes préparatoires") ~almost precisely over this: I felt like I was being taught physics like it was an axiomatic system where the tricks should not be questioned, they just work so "shut up and calculate" (and you don't even need to be doing quantum mechanics for that).
I just felt like we never got to the heart of the matter of why the models work and how to approach developing them, it was all about learning a bag of tricks.
Meanwhile, math and CS being a lot more axiomatic by nature, they also made a lot more sense to me.
That being said, that specificity of physics, the unbridgeable gap between reality and the models we build to describe it, in retrospect, is what makes it more interesting to me today (it's not just a "closed" system in the sense that math is — of course the relationship between math and physics is itself fascinating but that's yet another topic), but I still feel like I haven't found the right pedagogical approach to make it fit my mindset.
Your issue with physics but not with math reminds me a little of Hume's law. The difference that has always made that difference "make sense" to me is that math rules, even the axiom we use, are entirely chosen by the people using them, but the rules of physics are only useful if they match/predict what happens in the real world. For math we get to pick the ones that happen to be useful at a given time for a given problem (my go-to example of "it's all made up and the points don't matter" is why 1 isn't considered prime). For physics we're constrained to pick what best describes the real world. It probably helped that nearly all the physics course I had in high school/university had lab components focused on experimentally validating those rules/using those rules to predict results.
I think what it boils down to is that in my experience physics education lacks a clear historical component about how the current state of the art is a gradual process of proposing new models and rejecting old ones and figuring out the gaps between reality and the model. Instead, it feels like a God-given set of equations (that lots of people consider "the truth" for some reason), that you apply to cookie-cutter problems you must learn by rote. Though I understand the practical concerns (but then let's call it "physics for engineering"), as far as I'm concerned, you couldn't treat physics in a worse way.
The world just is, regardless of what we think about it. Physics is our best attempt so far to understand and predict it at a low level, but it will always be incomplete.
Maths (and especially compsci!) are constructions by and for humans.
Is it any wonder it is as you describe? It would be odd if it was any other way.
My point is precisely that I was often taught physics as if it was mathematics, where there is in fact a profound ontological difference between the two.
That's definitely an important point to consider, in fact something I think everyone in these conversations should be cognizant of, and also why it makes me believe the actual conversation should move to whether the device improves false positives/negatives rates or not (or at least has a chance to), which then might warrant wider access/use.
A better question is if people are going high-res, why not go high-res with tests whose accuracy is known, and for which there are useful, data-driven treatments?
Instead of casting a net of unknown quality every month, comparing against a null dataset (there does not exist a large dataset of these scans with outcomes for given markers).
Why not advocate cheap, easy blood/urine tests with higher frequency? Those tests do have large reference datasets with outcomes. And they have prescriptive value: there is likely more benefit to catching hypertension or diabetes earlier in more people.
