ClimateMachine.jl VS BlockBandedMatrices.jl

There are some big projects using all of this. The CLIMA climate model is a good example, and you can find 20 other fluid dynamics simulators. There's differentiable N-body simulators, rigid body and robotics simulators, DFT libraries, etc.

This isn't quite true. Differently ordered operations can lead to different results due to floating point math, but in general, the point of multiple dispatch isn't to produce different results for different types. The example from Stefan's talk does this when contrasting with operator overloading because it's easier to show a difference in results than a difference in performance.

I don't know the specifics of how Matlab/Octave do this, but you're probably right about runtime checks.

It's true that compilation time will make Julia slower than Octave for a single multiplication, but if you are doing a bunch of them (especially if they are small), Julia can perform the computations with much lower overhead since it doesn't have to do type checks for stable programs. Also, Julia lets you define specialized matrix types which can be asymptotically faster than Octave for specific circumstances. A great example of this is BandedBlockBandedMatrix from https://github.com/JuliaMatrices/BlockBandedMatrices.jl which are extremely useful solving of PDEs quickly, but aren't implemented in any of the other languages because they have to write all their dispatch systems manually for each algorithm.

Of course, (GPU-based) numerical linear algebra, complex numbers, etc. is all there. There's a bunch of fancy special matrix libraries for faster PDE calculations that can be mixed with all sorts of solvers, etc.