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Nothing reveals truths more effectively than patterns. If you can identify a pattern, you can see what actions conceal, what statements leave unsaid, and what agreements postpone. Peace treaties and the negotiations surrounding them are rarely analyzed in this way, even though they may exhibit patterns that recur across different circumstances. Consequently, there is often a simplification in understanding the pressures, incentives, and constraints shaping each step and each party’s choices, an exaggeration in linking the various warring factions into a single understanding without seeing the gaps, or a disregard for realities that might prevent us from truly understanding the

The post The Peace That Redistributes War: What Camp David Reveals About Lebanon and Iran appeared first on War on the Rocks .

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Boox Go 10.3 Gen II Lumi: One-minute review

As far as epaper writing tablets go, Boox has some of the most versatile options thanks to their Android-based operating systems, and the Boox Go 10.3 (Gen II) Lumi is arguably the best of them despite its rather mouthful of a name — I’m just going to call it the ‘Lumi’ from here on in.

I was already a fan of the original model, calling the reMarkable dupe a pleasure to use in my Boox Go 10.3 review — and by just adding a light and making the battery consumption more efficient, the Chinese ereader maker has made the Gen II model better.

Despite the screen being a touch slower than I’ve experienced with other devices like the 2024 and 2025 Kindle Scribes, the light adds a lot of contrast to the screen, making it easier to read on, whether that’s an ebook or a handwritten note. Now, I have to mention that there is a Gen II without a frontlight as well but, in my opinion, having the screen lit up just makes the epaper writing tablet that much more usable in any situation. With a frontlight, it also becomes a true Kindle Scribe alternative — again, thanks to its Android-based OS that gives the user access to the Google Play Store.

However, the aforementioned slow display is a disappointment as every time I woke up the Lumi to read or write, there would be a couple of screen refreshes (clearly marked as ‘Rendering’ when directly waking up to the native library application) before it settled down. Page turns, though, are fine but, again, Amazon’s 2025 Kindle Scribes are just that little bit faster at this as well. Long story short, screen performance doesn’t seem to have improved over the Gen I Go 10.3.

While I can live with the slower screen, I can’t fathom why Boox would retire the stylus that came with the original Go 10.3 as it was a lot better than the supplied InkSense Plus pen. The new stylus isn’t as smooth to write with, and is heavier, detracting from the all-important writing experience.

So as much as I’d love to say the Lumi is an improvement over its predecessor — and it is in a couple of ways — it just seems like Boox has taken two steps forward (with the frontlight and battery life) but one step back with the stylus.

(Image credit: Sharmishta Sarkar / TechRadar) Boox Go 10.3 Gen II Lumi review: price & availability

Announced March 2026

List price: $449.99 / £429.99 / AU$729

Available directly from the Boox Shop and third-party retailers

With the addition of the frontlight, the price of the Boox Go 10.3 (Gen II) Lumi is a lot higher than the launch price of the original Go 10.3 — you’re looking at shelling out $449.99 / £429.99 / AU$729 for the former, which is about $70 / £50 / AU$120 more than what the latter was when it was released in 2024.

That’s quite steep, although considering how expensive every bit of tech is getting thanks to RAM shortages, I’m not really surprised. It’s still competitively priced considering the 3rd-generation Amazon Kindle Scribe (2025 release with frontlight) retails for $499 / £449.99 / AU$849 for the 32GB flavor, while you get double the storage with the Lumi. Throw in the versatility of Android and it’s arguably better value (even compared to any of the reMarkables), but given its overall performance and user experience compared to the competition, I’d be hesitant to recommend it.

If the frontlight isn’t important, you can opt for the Gen II version without it, which will set you back $419.99 / £399.99 / AU$689. Prices for both Boox Gen II devices include the InkSense Plus stylus and a magnetic sleep case.

• Value score: 4 / 5

(Image credit: Sharmishta Sarkar / TechRadar) Boox Go 10. Gen II Lumi review: specs

Display type:

Undisclosed E Ink Carta HD panel

Screen size:

10.3 inches

Resolution:

300ppi

Processor:

2GHz octa-core Qualcomm Snapdragon 690

Frontlight:

Yes (warm and cold)

Storage:

64GB (non-expandable)

Battery:

3,700mAh

Speaker:

Dual (stereo)

Water protection:

None

Software:

Android 15

Connectivity:

Wi-Fi (2.4GHz + 5GHz); Bluetooth 5.1 USB-C

File support:

20 document, 4 image, 2 audio

Dimensions:

235 x 183 x 4.8mm (9.3 x 7.2 x 0.19 inches)

Weight:

364g (without case and stylus)

Boox Go 10. Gen II Lumi review: design & display

Good-looking, lightweight epaper tablet, but clunky stylus

10.3-inch E Ink display with good contrast

Textured rear panel adds good grip

With its eggshell-white body and metallic trim, I thought the first-gen Boox Go 10.3 was a pretty tablet, so I’m glad to see that the Gen II version hasn’t been changed — except for a slightly more textured rear panel that affords a little more grip, which is important for a slim device like the Go series slates. I should note here that both Gen II Go 10.3 tablets are identical physically but, without LEDs on the unlit version, it’s about 14g (0.5oz) lighter.

While the slimness doesn’t make for fantastic ergonomics, the Lumi weighing only 364g is, in fact, lighter than the Gen 1 model that tipped the scales at 375g without screen lights. It is about 0.2mm thicker than the previous Go 10.3, but that’s neither here nor there.

It’s still a beautiful device and, despite the thinness, there are dual speakers on either side of the USB-C port on the bottom edge, as does a mic. The only button is for power/sleep on the top edge. There’s no microSD tray here, though, for additional storage. One larger bezel allows plenty of room for your hand, although holding such a slim device, no matter how light, isn’t necessarily ergonomic. If you add in the sleep case, however, it just becomes heavier.

Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar I should talk about the case here, really quick — it’s a faux suede cover that unfortunately comes with the removable flap that I utterly detest. The brown suede material starts to lose its luster within days of use, so I can’t attest to its longevity, but I can’t fathom why Boox has to make the magnetic flap removable. This has been the case with the last few Boox devices I’ve tested and I find it rather annoying — the magnet is quite strong and the flap tends to entirely come away when you unfold it to close the case or secure the stylus to the side of the tablet.

(Image credit: Sharmishta Sarkar / TechRadar) Speaking of which, the magnetic hold of the stylus is good but, as with nearly every such device, it’s not strong enough to keep the pen secure during travel, so you do need to depend on the case.

Boox has also changed the pen, now offering the InkSense Plus stylus that has a thicker tip that just isn’t as smooth as the previous stylus. It’s also slightly heavier than the older pen, which makes it feel clunky in the hand that, in my opinion, doesn’t lend itself well to controlled mark making on the screen. And as I’ve discussed elsewhere, the stylus can make or break the user experience for epaper tablets . Sadly Boox is moving away from using EMR support on its screens (just like reMarkable), so you can’t use a third-party pen on the Lumi.

Boox hasn’t revealed what E Ink screen technology it’s using here, but the 10.3-inch display isn’t bad, with a 300ppi resolution. It’s not as yellow (warm) as I’ve seen on other Boox devices and the frontlight’s incremental adjustments for both brightness and temperature are excellent, adding wonderful contrast to the text or writing on the screen that makes the Lumi a far better reading tablet than its predecessor.

• Design & display score: 4 / 5

(Image credit: Sharmishta Sarkar / TechRadar) Boox Go 10. Gen II Lumi review: Software

Runs a custom Android 15-based operating system

A relatively clean user interface, but it takes getting used to

Plenty of good native apps, but access to the Google Play Store makes it very versatile

Where the original Go 10.3 released in 2024 was using Android 12, the Lumi (and its cheaper sibling) have been upgraded to Android 15, although the software is a very trimmed-down custom version of Google’s operating system. You still get unfettered access to the Google Play Store, so you can, in theory, use any Android app, but e-ink tablets weren’t meant for streaming content or playing games on.

That said, nearly every Boox device comes pre-loaded with native apps that are quite good in their own right and have been improved over time as well, making them more streamlined than before. That’s not to say the UX is easy — if you’ve never used a Boox device before, it will take time to wrap your head around it all as there are still little adjustments that are hidden in sub menus.

What’s different with the Go series compared to other Boox tablets is the home screen navigation — the main menu is arranged vertically on the left, with the Library being your default view. You can change the main menu items in Settings, though, either adding or removing applications.

Customizing the user experience has never been Boox’s problem, which means you can choose between several options for System Display or Gestures, and even tweak individual native apps as you see fit — even the NaviBall is optional to use and can be configured to trigger a plethora of functions.

(Image credit: Sharmishta Sarkar / TechRadar) The Control Center hasn’t changed much over the last couple of years, but I’m still not sure access to an AI assistant is necessary on an e-ink device, but perhaps some power users would appreciate it.

As always, you can sign into a fair few cloud services, including Google Drive and Dropbox, get full support for EverNote and OneNote, or download any other productivity application you might need, but keep in mind that not all third-party apps will be optimized for this device.

One of the things I really like about Boox’s premium note-taking tablets is OTG (on the go) support. This means I can plug in an external hard drive or portable SSD directly into the USB-C port to transfer files. In fact, it’s what I predominantly use even though BooxDrop is a handy app for cloud transfer. I don’t even bother signing into my Google Drive account on a Boox these days.

There really isn’t very much to complain about from a software standpoint — it’s the writing experience that takes a tumble here.

• Software score: 4.5 / 5

(Image credit: Sharmishta Sarkar / TechRadar) Boox Go 10. Gen II Lumi review: User experience

Good screen contrast with lights makes for a lovely reading experience

The supplied pen, however, isn’t as nice as the older version for writing

A device might look lovely and come with the nicest software, but ultimately its user experience determines its value. And that’s the case with the Go 10.3 Gen II Lumi — it does a lot right, like its reading experience is lovely, but its main functionality of writing is a little bit of a letdown.

Reading

Like I said, reading on the Lumi is lovely — the frontlight makes the text stand out, adding contrast where previously, on the Gen 1, it was disappointing. Page turns are quick enough too with the screen quite responsive to taps and gestures.

And yet I feel it’s a little slow for an epaper tablet that was released in 2026. For example, every time I woke up the Lumi during my months-long testing, it took a while to ‘render’ an ebook from its native library app.

Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar It very much seemed like it was refreshing the app twice before finally rendering the page. On the other hand, the 3rd-generation Kindle Scribe released in 2025 takes a second or two to open but doesn’t flicker or ‘refresh’ the page — it’s a lot smoother on waking up. As are the Kobo Libra Colour and Kindle Paperwhite from 2024 .

Boox doesn’t reveal what E Ink screen tech its using, but other than its slight slowness on waking up, I can’t complain about the reading experience. All books and images rendered on screen have been sharp and clear, particularly with the frontlight on.

Writing

The same kind of contrast with the frontlight on is available on the Notes app too where handwriting looks sharp and clear on screen. However, the physical act of writing on the Lumi isn’t as good as it has been on the Gen 1 because Boox decided to change the stylus.

The previous pen had a thinner nib that moved over the screen quite well. The new InkSense Plus stylus is heavier and the slightly thicker tip feels like it needs to be dragged to make a mark — it feels clunkier in comparison.

Boox has also stepped away from using EMR screens, which means you can’t swap that pen out for another. I tried using one of Boox’s own older pens (used to be supplied with the Note series) and it didn’t work. That’s a real shame as writing is the Lumi’s principal function and the experience of doing so is disappointing.

Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar The native Notes application, though, is full featured — in fact, for the average user it’s probably overkill — and would be ideal for students, teachers and researchers.

• User experience score: 3.5 / 5

Boox Go 10. Gen II Lumi review: Performance

Generally good performance except on waking up

A lot of ghosting visible, especially in the Notes app

Exceptional battery life for an Android device

The Lumi's overall performance is a mixed bag. While it uses a better chipset (Qualcomm Snapdragon 690) compared to the original Go 10.3 (Snapdragon 680), it hasn't really made much of a dent in the speed with which the Lumi performs the tasks at hand.

That's not a complaint — it's an observation. However, Amazon's newest Kindle Scribes are just that much faster at things, like page turns are so quick they barely register that they've moved and the lag between pen input and the mark on screen is barely discernible. On both counts the Lumi is just that smidgeon slower that you can notice if you use it side by side with the 2025 Kindle Scribe (3rd generation).

The slow waking up of the screen is also something that has bothered me, and I've already mentioned how disappointing the writing experience is, but I'm also surprised at how much ghosting there still is on Boox devices, including the Lumi, something Amazon has managed to completely eliminate on its Kindles. You'll be able to notice the ghosting on nearly every single image I have in this review (including in the gallery below).

Sharmishta Sarkar / TechRadar Sharmishta Sarkar / TechRadar Where the Lumi really shines (pardon the pun) is with battery life. It still uses the same 3,700mAh battery pack that was in the Gen 1 model and yet whatever software adjustments Boox has made has pushed battery life to weeks rather than days. It's the first Android digital notebook I've used to have such an impressive battery life, one that can match the Kindle Scribes, albeit dependent on use.

In my case, an hour of reading and 30 minutes of writing per day gave me 17 days of use with frontlight at medium brighness and a similar warmth setting, plus Wi-Fi on at all times (but not Bluetooth). Using cloud services and didn't make any dent on the battery usage, although excessive online browsing did drain a little extra but not by all that much.

Topping up was average, with the 3,700mAh pack going from 19% to full in just under two hours when plugged into a wall charger via a high-quality USB-C cable.

• Performance score: 3.5 / 5

(Image credit: Sharmishta Sarkar / TechRadar) Should I buy the Onyx Boox Go 10.3 (Gen II) Lumi

Attribute

Notes

Score

Value

It's cheaper than the 3rd-gen Kindle Scribe, but you'll get better performance elsewhere

4 / 5

Design & display

It's a beautiful device with reMarkable 2 vibes, and a frontlight makes a world of difference. However, the display just seems to lack the oomph factor I've seen in newer epaper tablets.

4 / 5

Software & apps

With an Android-based software allowing for versatility, and a nicer user interface than previous Boox UIs, it's hard to fault the Lumi. It's not perfect, but it ain't bad.

4 / 5

Writing & productivity

It's fantastic to read on, with brilliant contrast when the light is on, but the supplied pen is a major letdown when it comes to writing.

3.5 / 5

Reading

Without a frontlight, it's not the ideal ereader

3.5 / 5

Performance

On its own, the Lumi performs well for most users, but it's hard to give it full praise when there are faster E Ink displays being used on competitive digital notebooks, but the battery life is the best for an Android epaper tablet.

3.5 / 5

Buy it if...

You want a thin, light digital notebook with frontlight

There are more epaper writing tablets without a frontlight now than with, and trust me when I saw that can make a difference to usability, especially at nighttime and in bright sunlight.

You want the versatility of the Google Play Store on your digital notebook

It's not the only Android-toting epaper writing tablet, but it's well priced compared to some of the competition.

Don't buy it if...

You'll be doing a lot more writing than reading

The Lumi's main function is as a writing tablet and, given the disappointing stylus supplied with it, it's hard to recommend when there are competitors offering a better experience.

You want the best performance out of your digital notebook

It's not that the Lumi is bad; it's just that there are better options out there that have more responsive screens with next-to-no ghosting.

You need plenty of storage

64GB is a very good amount of storage that you'll get on the Lumi, but power users with large files saved on the device might need more, and there's no microSD card tray here. For that, I'd recommend Boox's Note series over the Go.

Also consider

I like the Lumi, but I don't I love it. If, like me, you'd be keen to check out other options, the two below are the ones I'd recommend, both of which are cheaper but aren't Android based.

Amazon Kindle Scribe (2nd gen)

It might have been succeeded by the 3rd-gen Scribe that's bigger and better, but it's still available to buy and still makes for an excellent e-ink writing tablet with a lovely screen and, importantly, regularly discounted now since it's an older model.

Read our full Amazon Kindle Scribe (2024) review for more details

reMarkable Paper Pure

It's not as full featured as an Android-based epaper writing tablet, but this reMarkable offers you a distraction-free slate to unleash your creativity. It's essentially the reMarkable 2 in a new body with 32GB of storage.

Read our in-depth reMarkable Paper Pure review to learn more

How I tested the Onyx Boox Go 10.3 (Gen II) Lumi

(Image credit: Sharmishta Sarkar / TechRadar) I've tested a range of ereaders and epaper writing tablets for TechRadar, so I'm well suited to compare the Boox Go 10.3 (Gen II) Lumi with its competitors.

As with all such devices I've previously tested, the first thing I did with the Lumi was to sign into the Google Play Store and down the Kindle and Kobo apps. I then used OTG support to transfer ebooks to the device directly from a portable SSD via the USB-C port, but I also used BooxDrop and cloud transfer from Google Drive to test various services.

The Lumi then became my daily ereader and note taker for 4 weeks, using it an average of 2-3 hours a day to read and write. I'm not an artist, so drawing was out of the question.

Out of curiosity, I also downloaded Geekbench 6 to benchmark the CPU (the app is available on the Play Store). I sideloaded a few MP3 files to test the onboard speakers.

Read more about how we test

[ First published July 2026 ]

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Beyond the limits of the theoretical physics we currently understand, many of the most popular ideas have one thing in common: at their core, they’re mathematical extensions of what we presently know and understand. By adopting a mathematical framework that not only encompasses what we presently understand it, but that seeks to explain more things than our currently prevailing theories do, we hope to extend our understanding of the Universe past the current frontiers. We know that our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don’t — and can’t — do everything.

Our theory of gravity and our quantum theories of particles and fields are fundamentally incompatible with one another: they cannot explain, for example, how the gravitational field of a quantum system behaves. Moreover, our current framework cannot sufficiently explain:

dark matter,

dark energy,

the ultimate origin of our Universe,

or the reason why our Universe is filled with matter and not antimatter,

among other puzzles. It’s true that mathematics enables us to quantitatively describe the Universe; it’s probably the most useful tool we have when it’s applied properly. But the Universe is a physical, not mathematical entity, and there’s a big difference between the two. Here’s why mathematics alone will always be insufficient to reach a fundamental theory of everything.

One of the great puzzles of the 1500s was how planets moved in an apparently retrograde fashion. This could either be explained through Ptolemy’s geocentric model (left), or Copernicus’ heliocentric one (right). However, getting the details right to arbitrary precision was something neither one could do. Both models have little predictive power; they could not detail the orbital properties of a hypothetical additional planet the way a more concrete physical theory, like Newtonian or Einsteinian gravity, would later do.

Credit : E. Siegel/Beyond the Galaxy

About 400 years ago, a great existential battle was unfolding about the nature of the Universe. For millennia, astronomers had accurately described the orbits of the planets using a geocentric model, where the Earth was stationary and all the other objects orbited around it. Armed with the mathematics of geometry and precise astronomical observations — including tools like circles, equants, deferents, and epicycles — the precise mathematical description of the heavenly bodies’ orbits matched what we saw spectacularly. For over a thousand years, this Ptolemaic model was largely undisputed for an underappreciated reason: because it was so much more successful, accurate, and had greater predictive power than any alternative.

The match between theory and observation wasn’t perfect, however, and attempts to improve upon it were frequently made. In most cases, this led to the addition of more epicycles, and then in the 16th century, the idea of Copernicus’ heliocentrism came about. By placing the Sun at the center, explanations of retrograde motion — as illustrated above — became simpler, but the fits to the data were worse than in the geocentric model. When Johannes Kepler first came along in the 1590s, he had a brilliant idea that sought to solve everything.

Kepler’s original model of the Solar System, the Mysterium Cosmographicum, consisted of the 5 Platonic solids defining the relative radii of 6 spheres, with the planets orbiting around the circumferences of those spheres. As beautiful as this is, it couldn’t describe the Solar System as well as ellipses could, or even as well as Ptolemy’s model could.

( Credit : Johannes Kepler, 1597)

He noticed that, on an astronomical front, there were six planets total, if you included Earth as a planet, but not Earth’s Moon. He also noticed that, mathematically, there were only five Platonic solids: five mathematical objects whose faces are all equal-sided polygons. By drawing a sphere inscribed inside and circumscribed outside each one, he could “nest” them in a way that fit the planetary orbits extremely well: better, honestly, than anything Copernicus had ever done. It was a brilliant, beautiful mathematical model, and arguably the first attempt at constructing what we might call “an elegant Universe” today.

But observationally, it was still a failure. It failed to even be as good as the ancient Ptolemaic model with its epicycles, equants, and deferents. The idea was one of brilliance, elegance, and simplicity, and marks our first attempt to argue — from pure mathematics alone — how the Universe ought to behave. But when we confronted the idea with reality, and the actual observations of the planets, it simply didn’t work. The prevailing model at the time, with all of its relative ugliness, still matched reality better than Kepler’s beautiful model did.

So what did Kepler do next? In many ways, he engaged in a stroke of genius that would ultimately define his legacy.

Even before we understood how the law of gravity worked, we were able to establish that any object in orbit around another obeyed Kepler’s second law: it traced out equal areas in equal amounts of time, indicating that it must move more slowly when it’s farther away and more quickly when it’s closer. At every point in a planet’s orbit, Kepler’s laws dictate at what speed that planet must move.

Credit : Gonfer/Wikimedia Commons, using Mathematica

He took his beautiful, elegant, compelling model that he had spend so long developing — a model that that he clearly loved — and, because it disagreed with observations, he did the most remarkable thing of all: he threw it away. Instead of working to refine and massage his model to better match what we saw, he dove into the data to find what types of orbits would match how the planets were actually seen to move. In the aftermath of his analysis, he came away with a set of three profound scientific (not purely mathematical) conclusions.

All six of the planets, despite assumptions, didn’t actually move in circles around the centrally located Sun, Instead, those planets moved in ellipses with the Sun at one focus, with a different set of parameters (like orbital speed and orbital eccentricity) describing the ellipse of each planet.

Because their orbits were non-circular, planets didn’t move at a constant speed in their orbits around the Sun. Instead, each planet moved at a speed that varied with the planet’s distance from the Sun — with faster speeds corresponding to closer approaches to the Sun and slower speeds corresponding to greater distances from the Sun — in such a way that each individual planet sweeps out an equal area over the same interval of time.

And finally, planets exhibited orbital periods that were directly proportional to the long axis (the major axis) of each planet’s orbital ellipse, their distance from the Sun, raised to a specific power (quantitatively determined to be 3/2).

This animation shows the four super-Jupiter planets directly imaged in orbit around the star HR 8799, whose light is blocked by a coronagraph. The four exoplanets shown here are among the easiest to directly image owing to their large size and brightness, as well as their huge separation from their parent star. Our ability to directly image exoplanets is constrained to giant exoplanets at great distances from bright stars, but improvements in coronagraph technology will dramatically change that story.

Credit : Jason Wang (Northwestern)/William Thompson (UVic)/Christian Marois (NRC Herzberg)/Quinn Konopacky (UCSD)

These three laws represented a pivotal moment in the history of science. It wasn’t philosophy, theology, or even mathematics that was at the root of the physical laws that governed nature. We couldn’t divine the “most perfect” mathematics that described nature, but instead could only use mathematics as a tool that describes how the physical laws of nature manifest. The key advance that happened was choosing to root our scientific models for reality in what was observable and measurable, and choosing the model that best fit our collected data.

This was the start of our since-unbroken tradition, for science, that any theory must confront itself with data from the Universe itself. Without this idea, further progress becomes impossible.

This idea has come up again and again throughout history, as new mathematical inventions and discoveries continuously empower us with new tools to attempt to describe physical systems. But each time, it wasn’t that the new mathematics told us how the Universe worked. Instead, new observations told us that something beyond our currently understood physics was required. Pure mathematics alone is insufficient to lead us to the next advance, but if we’re lucky, some of our already-discovered mathematical ideas will indeed wind up successfully describing the Universe we observe.

We often visualize space as a 3D grid, even though this is a frame-dependent oversimplification when we consider the concept of spacetime. In reality, spacetime is curved by the presence of matter-and-energy, and distances are not fixed but rather can evolve as the Universe expands or contracts. Prior to Einstein, space and time were thought to be fixed and absolute for everyone; today we know this cannot be true. If you place a particle on this grid and allow the Universe to expand, the grid will expand, too, and hence the particle will appear to recede away from you.

Credit : Reunmedia/Storyblocks

We can fast-forward to the early 1900s: when it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, which was better described by Einstein’s special theory of relativity. Newton’s theory of universal gravitation was in similarly hot water, as it could not explain the observed motion of the planet Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like the 3D grid shown above) had been floating around for decades among mathematicians.

Unfortunately, developing a mathematical framework to describe the possibility of curved spacetime (and gravitation) required more than pure mathematics, but the application of mathematics in a particular, tweaked way that would agree with observations of the Universe. There are many possible mathematical ways to write down equations that govern a curved manifold, but only one of those mathematical possibilities — at most — can correspond to the spacetime that describes our Universe. This, quite possibly, is the main reason why nearly everyone on Earth know the name “Albert Einstein,” but hardly anyone knows the name “David Hilbert.”

Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been ‘straight’ lines to instead become curved by a specific amount. In general relativity, we treat space and time as continuous, but all forms of energy, including but not limited to mass, contribute to spacetime curvature. In addition, the distances between unbound objects evolve with time, owing to the expansion of the universe.

Credit : Christopher Vitale of Networkologies and the Pratt Institute

Both men — Einstein and Hilbert — had done substantial work on theories that linked spacetime curvature to gravity and the presence of matter and energy . They overlapped in location, spoke to each other, shared research ideas, and were quite fond of one another. Both of them had worked with similar mathematical formalisms; today one of the most important equations to come out of General Relativity is known as the Einstein-Hilbert action . But Hilbert, who had come up with his own, independent theory of gravity from Einstein, pursued bigger ambitions than Einstein: his theory attempted to include explanations for matter and electromagnetism as well as gravity, all in the same framework.

The problem was this: Hilbert’s ambitious theory simply didn’t agree with nature. Hilbert was constructing a mathematical theory as he thought it ought to apply to our physical reality, rather than seeking to let reality be the guide that led him to the equations that best described it. Because of Hilbert’s approach, now generally recognized as being a fundamentally backwards way to develop a theory, Hilbert never got successful equations out of his framework that accurately predicted the quantitative effects of gravity.

Einstein’s General Relativity, on the other hand, quite famously did, and that’s why the field equations are known as the Einstein field equations, with no mention of Hilbert at all. Unless your idea can survive a confrontation with reality, you aren’t doing physics at all.

The wave pattern for electrons passing through a double slit, one-at-a-time. If you measure “which slit” the electron goes through, you destroy the quantum interference pattern shown here. Regardless of the interpretation, quantum experiments appear to care whether we make certain observations and measurements (or force certain interactions) or not.

Credit : Dr. Tonomura; Belsazar/Wikimedia Commons

An almost identical situation would come up again just a few years later in physics history: this time, in the context of quantum physics. No matter how meticulously you set up your experiment, if you were to fire an electron through a double slit, it was impossible to know, based on the setup and its initial conditions, where it would wind up. You could only know what the odds were of it landing along a particular distribution of possible locations. A new type of mathematics — one rooted in wave mechanics and a set of probabilistic outcomes — was required. Today, we use the mathematics of vector spaces and operators, described by a term that might ring a bell to physics students: Hilbert space .

The same mathematician who worked on the physics of spacetime alongside Einstein, David Hilbert, had also discovered a set of mathematical vector spaces that was enormously promising for quantum physics. And interestingly enough, just as was the case for Hilbert’s attempt at a theory of gravity, the predictions arising from his Hilbert space didn’t quite make sense when confronted with physical reality.

To get a match with physical reality, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. (Where the “inner product” of that Hilbert space has additional physical constraints placed on it, but not for any mathematically motivated reasons.) The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.

The pattern of weak isospin, T3, and weak hypercharge, Y_W, and color charge of all known elementary particles, rotated by the weak mixing angle to show electric charge, Q, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass. This diagram shows the structure of particles, but is rooted in both mathematics and physics.

Credit : Cjean42/Wikimedia Commons

Today, it’s grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality . A number of mathematical-based approaches have been tried over the years and decades, including to embed our currently understood reality into a grander framework by:

imposing additional symmetries,

adding extra dimensions,

adding new fields into General Relativity,

adding new fields into quantum theory,

or using larger groups (from mathematical group theory) to extend the Standard Model ,

along with many other scenarios.

Here’s the thing: mathematics is a powerful tool, but it is not a unique tool. When you have a mathematical framework, it generally admits many possible solutions. However, at the end of the day, there’s only one physical reality, and so you must impose something from physics — rules, constraints, selection effects, etc. — that allows you to pick out the one mathematical solution that corresponds to that observed, measured reality.

From a theorist’s perspective, the type of mathematical exploration that we can conduct are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what’s presently known. But mathematics alone cannot teach us how the Universe works. We will obtain no definitive answers without confronting its predictions with the physical Universe itself.

Visualizing the multiplication of the unit octonions, of which there are 8, requires thinking in higher-dimensional spaces (left). The multiplication table for any two unit octonions is also shown (right). Octonions are a fascinating mathematical structure, but offer non-unique solutions to a myriad of possible physical applications.

( Credit : Yannick Herfray (L); English Wikipedia (R))

This is not some lesson that only comes along at the frontiers or at the highest levels of theoretical physics. In one way, in fact, it’s a lesson that every physics student learns the first time they deal with projectile motion, and attempt to calculate the trajectory of an object thrown into the air. Questions you might want to answer include:

How far does the object go?

Where does the object land relative to your initial location?

How long does that object spend in the air before hitting the ground?

To be sure, there are mathematical equations, the equations of motion , that provide the answers. But when you solve those equations governing these objects, you don’t get “the answer.” You get several answers; that’s what you can arrive at from mathematics alone.

But in reality, there’s only one object. It follows one only trajectory, it lands in one and only location, and it does so at one specific, quantifiable time. Which of the possible answers that the equations give you corresponds to that reality? Mathematics not only won’t tell you, but it fundamentally can’t tell you. To get the answer that corresponds to reality, you need to understand the particulars of the physics problem in question, as that information is essential to determining which solution has a physical meaning behind it. Mathematics will get you very far in this world, but it won’t get you everything. Without a confrontation with reality, you won’t be able to tell the real world from worlds that are pure mathematical fantasy. If your goal is to understand the physical Universe, mathematics just isn’t enough on its own.

This article was first published in February of 2023. It was updated in July of 2026.

This article The physical Universe isn’t made of pure mathematics alone is featured on Big Think .