David Hilbert

So these are seven problems that were put forward by something called the Clay Math Institute, I think the year 2000.

And they were inspired by a mathematician named David Hilbert, who in 1900 had put forward 23 problems that he felt could shape the course of math in the 20th century.

And this was enormously influential.

Hilbert's list of problems.

Where does the energy from redshifted photons go? In 1915, the expansion of the universe hadn't yet been discovered, but the failure of energy conservation was still clear from the math of general relativity.

Two of the greats of the era, David Hilbert and Felix Klein, sought the help of a young mathematician, Emmy Noether, to understand this seeming paradox.

She discovered why the law of conservation of energy broke down in general relativity.

The law was not fundamental after all.

Some of them have to do with prime numbers, like I hinted at the beginning, but they come from all different topics.

And every now and then there was a famous list of problems in the year 1900 that this mathematician David Hilbert assembled.

And I forget even how many he listed, but a couple dozen maybe.

And these really helped motivate and guide the development of mathematics for the beginning of the 20th century.

So it directed our efforts.

It helped us know what we should work on and what was considered important and what would illuminate the dark spots of our knowledge that we most needed addressed.

Now we go another hundred years forward in the year 2000 and an institute with a lot of guidance from mathematicians, but also a lot of money from industry called the Clay Institute says, okay, we need, we're ready.

It's been a hundred years since Hilbert.

Yeah, we've had 15 years to prepare for this, but in the best traditions of not doing your homework until the night before, here we are turning around two episodes in record-breaking time.

Yeah, so we first heard about this project back in the January 2013 edition of Literally, when Neil and Chris revealed they'd spent much of 2012 working on a musical project telling the life story of British scientist, mathematician and codebreaker Alan

Turing.

Now, let's just spend a little while getting to know about Alan Turing, because I admit he's not someone I knew a whole lot about before he came on my radar, thanks to Pet Shop Boys.

And this is how ideas would form in his mind.

He was dreaming of machines.

In 1936, he published a paper in answer to a question posed by the German mathematician David Hilbert.

Can all of mathematics be proved by logic alone? In solving this problem, Turing imagined a device.

I mean, it's clear, it's clear that no interpretation, no, you know, interpretation of quantum mechanics can explain Bell's theorem in a completely satisfactory way.

So the, the, the picture I, um, or the, the line I, I took was to say, "Well, what could we do to modify quantum mechanics?" And, you know, what could be the weak points of quantum mechanics? And I was driven very much by, you know, people like David Hilbert actually, who, you know, Hilbert space is famously named after or was named by John von Neumann as the state space of quantum mechanics.

But, you know, Hilbert space is a, is a continuum space.

So, you know, it's, it's a mathematics, a vector space over the continuum field of complex numbers.

A great many mistakes were made by 17th and 18th century mathematicians because they didn't understand the importance of rigour.

It is just that in achieving rigor through magisterial projects like those of David Hilbert or the Burbaki School, mathematics became afraid of speculation and plea.

Mathematics books are devoid of humor, personality, and any indication that a human mind contemplated the results.

The texts are an endless succession of claims, proofs, and exercises.

But here's the problem.

If all of us in this world are like, you know, Spitzer's or all of us are like Einstein's for that matter or David Hilbert's for that matter or whoever.

So if we're all just super geniuses, but we still have a ton of questions that we cannot answer from within our own intelligence and systematic knowledge through our intelligence.

If that happens, well, gee whiz, can we be smart? you know, completely intelligible.

Instead, they chart an arrow pointing through a map of possible futures.

A 26 August 2026 essay in Quantum Magazine asks a surprisingly basic question about this mathematical arena known as Hilbert space.

For a century, scientists have used it to predict how matter behaves with perfect accuracy — Yet, they still cannot agree on whether this space is just a useful fiction or the actual hidden theater of reality.

Written by Charlie Wood in Quantum Magazine.

Narrated for ARVA.

In Hilbert space, all things are quantumly possible.

At the heart of quantum mechanics lie a few sacred rules for how to use the theory.

First and foremost is, roughly, that thou shalt not think about ordinary objects presently whizzing through ordinary space.

Hardy gave himself 25.

He gave Littlewood a 30, then he gave David Hilbert an 80 and Ramanujan a 100.

Whether those numbers were meant literally is beside the point.

Hardy regarded Ramanujan as possessing a kind of mathematical genius that was extraordinarily rare.