Why not? Exactly because there is nothing tethering our axioms on paper to what is necessarily true. You could formulate something wildly different from ZF±C/Peano/whatever normal axiom system, but we wouldn't call it "math", and what we currently call "math" will work under any conditions
Our 'math' will work under any of our conditions (as far as we can observe), but who's to say they can't have 'math' in another universe that will work under any of their conditions, yet still be different from ours?
That's what GP was saying ("there are no conceivable alternate universes with different math", and none with a different derivative in particular), but I see no reason why math-as-we-know-it couldn't just be inapplicable to different 'conceivable' universes.
With no mathematical rigor there is no mathematical understanding. You are robbing yourself, as the concepts are meaningless without the context.
Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true. All of this knowledge is free, and much more interesting than a formal restatement of "it was not supposed to rain, but I see clouds outside, I guess I'll expect light rain instead of a big thunderstorm".
> With no mathematical rigor there is no mathematical understanding. You are robbing yourself, as the concepts are meaningless without the context.
I will think more about this, but I'm not sure I agree. I have enjoyed reading Feynman talk about twins and one going on a supersonic vacation without understanding the math. Verisimilitude allows a modeling of understanding with a scalar representation of scientific knowledge, so why not?
Of course I would like to understand the math in its purest forms–just the same as I wanted to read 1Q84 in Japanese to be able to fully experience it in its purest form, but my life isn't structured in a way were that is realistic even if the knowledge of the Japanese language is free.
> Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true.
I can't even foil so the journey toward understanding can feel unattainable in the time resources I have. This absolutely may be a limiting belief, but the concept of knowledge being free ignores the time cost for some exploring these outside of academia or professional setting.
Indeed everything has an opportunity cost, and every life has its own priorities.
Since you mention Feynman, I would like to observe that many expositors who target the lay audience have the skill of making the audience believe that they have comprehended(1) something of an intellectual world that they have no technical grounding to truly comprehend(2). In my view these are two distinct types of comprehension/understanding. So long as the audience is clear on which type of understanding they are getting, and is not wasting time unwittingly pursuing one type at the expense of the other then I see no harm.
There is a risk however, that the pop expositors will put you in a headspace where even if you are faced with accessible, but type 2, material you will not be familiar with what really constitutes understanding. As a mature age student it took me quite a few years of maths exams to switch from 1 to 2. Nowadays I am more comfortable with admitting that I don't understand some piece of math (for that is the first step on the path to learning) than being satisfied with a pop-expository gist.
I've thought a lot about this exact topic. You need both to do well.
You need handwavy and vague versions of things to understand the shape of them and to build intuition.
Then you need to test the intuition and build up levels of rigor.
Especially in the context of the Kalman Filter. I just helped a bunch of middle school students build a system for field localization and position tracking. They don't have all kinds of knowledge. They don't have linear algebra or a real understanding of something being gaussian and have to have a bazillion variables. They understand that their estimates and the quality of stuff coming off their sensors have different qualities based on circumstances, and that gain needs to vary. They'll never hit the optimum parameters.
But: their system works. They understand how it works (even if they don't know how to quantify how well it works). They understand how changing parameters changes its behavior. When they learn tracking filters and control by root locus and all kinds of things later, they'll have an edge in understanding what things mean and how it actually works. I expect their intuition will give them an easier time in tackling harder problems.
Conversely, I've encountered a bunch of students who know what "multimodal" means but couldn't name a single example in the real world of such a thing. I would argue that they don't even know what they're talking about, even if they can calculate a mixture coefficient under ideal conditions.
There's a lot of fluffly language here that isn't saying much.
Linear algebra is not something that takes years of patient study to gain basic competency. It had almost no prerequisites and can be understood enough to understand least squares in a focused weekend or two.
Thank you for the encouragement. I'll will take a week or two and spend some time with some focused learning. Do you have any recommendations where to start?
> With no mathematical rigor there is no mathematical understanding
While I appreciate rigor to really know deep details, is not only not a requirement for understanding, but a hurdle. A terrible insurmountable hurdle.
To first have understanding, I need some kind intuition. Some explanation that makes sense easily. That explanation is btw, what typically the inventor or discoverer had to begin with, before nailing it down with rigor.
> Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true.
None of these are needed, or even useful, for understanding the Kalman filter.
I don't agree at all, vcv rack helped me understand synthesis in a much deeper way than I would have otherwise. What's a retrigger? Oscillator drift? Why do you modulate with a lfo? These are much simpler to understand when you're patching modules by hand in vcv, especially when you start with a blank slate.
On the other hand, before vcv, seeing a vst synth just had me overwhelmed instead.
I'd recommend everyone reading this to get free vcv + the surge vcv library, and just play around with it.
Gotcha; it's less that my complaint doesn't apply, but more that it isn't relevant (i.e., "squares" and "cubes" aren't especially interesting constructs which need to coexist nicely, and if you relax that constraint then directional geometry can be very interesting). Does that sound right?
- Investing provides benefits for society at large
- Investors are exploiting the labour of others for their own gain
(but also your examples only work in a very weird worldview where everything is privatised, but I don't want to bother discussing that on this website)
Not at all, pretty much all popular languages (except C/C++) are as safe as (safe) rust. The only safety rust brings to the table is memory safety, which most languages achieve with a runtime and a garbage collector, which have a performance tradeoff.
In three most narrow definition of safety I agree. But that's a very narrow definition. Rust does offer a lot more:
- no undefined behavior
- many classes of concurrency bugs prevented by the type system
- standard library and much of the ecosystem makes invalid states unrepresentable. E.g. a String is always valid UTF8
Those are things that are true to varying degrees for other languages. Dart does pretty well imho. But for example Java and C# offer memory safety but have very unsafe concurrency
Or you have "standard" vest (without ceramic armor plates) and the shooter shelled out for handgun designed for (and with) armor-piercing ammo or PDW-class gun or higher powered cartridge in a full rifle.. etc ;)
It's not, we just use NAND everywhere because they're easier to make with transistors. You can get functional completeness with a NOR instead, or alternatively with some different combinations of other logical operators.
We even implement AND gates with NANDs in electronics (because they're way simpler), but we might not have to limit ourselves to a single base gate with mechanical computers.
If you stop squinting, you'll see that neither the average exclusive activity enthusiast, nor a person using "normie" unironically are shallow nor superficial.
Maybe your experience can be explained by how you see these people, not by how these people actually are.
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