On the other hand, Michael’s argument about drag racing is compelling too. Just because I can do something unbeautifully on my current machine doesn’t mean that if I were clever enough I could do the same on the 8086 I first owned.
The tool used to move from powerful back to less-powerful is somewhat different in either case though. In drag racing, you’re hoping that better oil, improved rocket fuel, polishing the hub-cab, and changing the aerodynamics of the body - will make you go faster. In computing, you’re hoping that a mathematical insight will make you go faster.
They’re a theoretical similarity; perhaps some incredible new insight into aerodynamics will give a lawnmower a new body and make it go FTL.
That said, I think mathematical insights have a track-record of unsurpassed retro-applicable power.
Like bubblesorting for a century, and then suddenly coming up with quicksort.
I’ve seen a lot about people that’s simple enough to make me think you might be right.
Though the bits that aren’t make me wonder.
…make you wonder if there’s something “different”, or something that’s “a matter of time”?
Regarding mathematics’s powerful ability to retrospectively simplify - without it, the difference in growth rates over spaces of increasing dimension would have long been a cause for despair.
Consider the body of “human wisdom/knowledge”, our collective map or manual for reality. Intuitively, the map is a N dimensional subspace embedded in our universe, which is M >= N dimensions.
Some 6,534,658 dimensional blob that captures the fully recursive expansion of human beliefs that match reality. The space exists within a 42,378,943,278,942,378,748,239 dimensional space that represents the entire universe of true beliefs that intelligent beings can have about reality.
Perhaps one day both spaces will collapse into a single Awesomely Cool Grand Theory Of Everything That Is Awesome And Cool, but until such an “Omega-day” (now that’ll be a national holiday I’m sure) we’re mostly expanding our belief space.
In other words, our map of reality has many axis along which our wisdom can grow.
Nice.
Dimensionality reduction – on the dataset of “all data about the universe”.
For the sake of visualization, consider the blob a plain old fashioned 3D sphere that demarks the limits of human knowledge.
Two BC it was small… Then it grew steadily until 1100 AD, hovered in place a little during the dark ages… and the last two centuries it has grown enormously….
But remember I’m talking about the collective sphere. We also have our individual spheres.
When we’re born, we’re pretty much at zero size. We have a tiny built-in evolutionary grasp of reality - but that’s it. From then on, it’s up to us to grow the sphere.
And for those who dare, to extend parts of the personal sphere to the outer rim of the collective sphere. That’s where you can push on the surface and expand it.
If you’re not bored yet, then you’ll be happy to hear I can make my point at long last…
Actually, I’m trying to reconcile your picture with mine.
Namely, that traveling is largely a one dimensional effort
That is to say, from one Planck-moment to the other, you’re only going in one direction.
Right.
Hopefully, you can trace out 2D areas, sweep out 3D volumes, and have progress along one axes benefit higher dimensions…
Yeah, that’s the reconciliation there.
Now because our lives are finite (for the time being anyway), speed of light is a killjoy, and we have to eat and sleep, there’s an upper bound on the volume one can sweep out.
That is, until you leverage mathematical retrospect and abstract.
I saw on reddit the other day that a Royal Society had published a bunch of old papers on the net.
People get really excited about that, but if you want to see the fruits of the kind of simplification you’re talking about, then a comparison of, say, Schrodinger’s early papers on quantum mechanics and a recent textbook will do the trick.
(Not that it’s not still fun to read the old papers)
The notion of “traveling” from the moment your born towards the outer surface of the collective sphere (and beyond for some)…. I really enjoy that visualization
First, because it so nicely captures the time and distance it takes (standing on the shoulders of giants past) until you reach the edge - which is where you can push on the wall and go beyond it…
Second, because growth-rates increase non-linearly with dimension, we instantly recognize that we must always find ways to improve our travel speed.
Yeah. Also, the fact that we now spend 20 years in formal schooling to even come close to the edge is helpful. And it helps, for that, that we’re living longer.
Totally hear you on the Schrodinger papers… People internalizing highschool math as a bag-of-tricks notwithstanding, some subset of today’s 20-year-olds do internalize math as a language to describe reality to a point beyond Newton’s at age 40.
Sure. Language is power.
Frankly, I see teaching Physics as fundamentally a problem of teaching language. I mean, if you really knew what is meant by acceleration, wouldn’t it be immediately obvious that v = v_0 + at for constant acceleration?
It would be, though I wonder if the obviousness doesn’t fall from a circular kind of reasoning (through your implicit “really know what is meant by acceleration”).
I do agree that language is a root issue…
Not surprising, considering that language is often the most effective mechanism for communicating new ideas.