While multidimensional geometry could be regarded as a separate discipline, dimensions themselves does not exist. It is mere an abstraction created by an observer.
A sphere makes a perfect sense in a in this Universe, while a 4D hyper-sphere (and so-called hyper-cube) is an utter nonsense. Of course, one could describe such an abstraction in as a mathematical object, but it will be mere an abstraction, like Hegelian ones. It does not exist outside people's heads.
All the billions years of this planet never revealed a single dimension. All the geometry we have in proteins is based on the notion of 3D spheres - a notion of the same distance in all possible dimensions (given that there is no such thing as dimension).
Same logic could be applied to refute any string theory sectarian nonsense, given that just one single contradiction is enough. Non-existence of time as a phenomena is enough to destroy all the space-time curvature mathematical or rather Hegelian crap.
But who cares.
This is the discussion https://news.ycombinator.com/item?id=12616426
This is my comment:
Nothing to see here. In the age of hipsters, cosplay of intelligence and "social" role-governed behavior in general there is no surprise that we could read all that nonsense in the New Yorker, which is a hipster magazine.
No sane person, who have enough intelligence to realize that music, sports, arts and craftsmanship are 95% based on practice would doubt it. Any good music teacher or Olympic coach will tell you this. The 5% of talent determines where most likely would end up on the spectrum, but talent without practice is nothing, like knowledge unapplyed.
Even in programming actually writing code on the daily basis is what distinguish top performers from mere mediocrity and talking heads and bloggers at the bottom.
For the hipster narcissistic sub-culture it is perfectly OK to "challenge" (they think it is a challenge, not a display of stupidity and lack of even basic understanding) the nature of reality with sophisticated (which does not imply intelligent) nonsensical blah-blah-blah in order to gain public attention to their acting according to the role.
Hipsterism as a social phenomena is an insult to intelligence the way commercialized empty yoga of cosplay and asanas is an insult to spirituality (which is seeking for the truth).