Lost in Math?

I finally got around to reading Sabine Hossenfelder’s ‘Lost in Math‘ (2018).

It fully deserves its praise. The book is, as the reviewers write, accessible, well-informed, and engaging—at times even genuinely funny. The structure, built around interviews with leading theorists, gives it both breadth and credibility. It is, without doubt, one of the better popular accounts of modern theoretical physics.

It also felt familiar.

Hossenfelder and I belong to roughly the same generation. As teenagers in the 1980s, we were fascinated by the same questions: What is the Standard Model really about? Where did it come from? What problems did it solve that even Albert Einstein or Max Planck could not? And what new questions did it open?

And then, of course, the next layer: why do we need theories beyond it—string theory, supersymmetry—if the Standard Model already works so well? What are these theories trying to explain that the Standard Model cannot?

And what should we make of the experimental side of things? From the discovery of the Higgs boson to the evidence for dark matter, dark energy, and gravitational waves—what do these findings actually mean?

Hossenfelder chose to pursue these questions within academic physics. I did not. I studied economics, but continued to explore physics as a personal project—especially after 2012, when the Higgs boson was announced. By then, I had grown dissatisfied with popular science accounts and felt the need to understand the mathematics itself.

And yet, after working through the math, I found myself asking a different kind of question: not whether the equations work, but what they mean.

It is here that Hossenfelder’s book, for me, remains incomplete.

Beauty, Truth—and Something Missing

The central argument of Lost in Math is well known: modern theoretical physics has been led astray by an overreliance on aesthetic criteria—symmetry, elegance, mathematical beauty—at the expense of empirical grounding.

That critique is compelling, and I largely agree with it.

But it seems to stop halfway.

While Hossenfelder questions the role of beauty, she does not fundamentally question the underlying framework itself. The Standard Model and its extensions remain, in her account, the unquestioned language in which physical truth must ultimately be expressed.

What is largely absent is a deeper discussion of physical interpretation.

The Question of Meaning

Let me be more concrete.

The book does not attempt to explain why the strong force could not be understood in more classical terms, for example as some form of electromagnetic interaction arising from internal charge dynamics.

It does not address why abstract quantum numbers—color charge, flavour, isospin—should be regarded as physically compelling, rather than as mathematical constructs that work but lack intuitive grounding.

Likewise, the weak force appears mainly as part of a formal structure, without much discussion of what it might represent in more tangible terms—such as the distinction between stable and unstable particles.

And perhaps most strikingly, the book does not engage in any depth with the meaning of the most fundamental relations in physics: the quantization expressed in the Planck relation, or the significance of mass-energy equivalence. These are presented as known facts, not as conceptual puzzles.

None of this is a flaw in the usual sense. It is simply not the book Hossenfelder set out to write.

But it is the book I was hoping to read.

Old Physics, Reconsidered

So where does that leave us?

In my own work, I often find myself returning to what many would call “old physics”: Maxwell’s equations, together with relations like Planck–Einstein relation and mass–energy equivalence.

This may seem old-fashioned. Perhaps it is.

But I am increasingly convinced that the real challenge is not to extend the mathematical formalism, but to understand what the existing formalism is telling us about physical reality.

From that perspective, the problem is not only that modern physics may have followed beauty too far. It is also that it may have drifted too far from meaning.

A Different Kind of Dissatisfaction

Hossenfelder ends her book on a note of optimism. Physics, she argues, will continue to make breakthroughs, and those breakthroughs will—once again—be beautiful.

I hope she is right.

But closing the book, I was left with a different thought. Not frustration, but a kind of clarity.

I realized that I am quite content continuing to explore these questions from a more classical, more intuitive starting point—even if that places me outside the mainstream.

Because, in the end, the question that still matters most to me is a simple one:

Not whether the mathematics works, but whether we truly understand what it is saying.

Post scriptum on the 2019 revision of SI units

Sabine Hossenfelder finished and published her book in 2018—just before the 2019 revision of the SI units.

I find myself wondering whether that revision is, in its own quiet way, more meaningful than many of the theoretical developments discussed in her book. Perhaps I am over-interpreting, but this is how it looks to me.

The revised SI system fixes exact numerical values for a small number of fundamental constants, such as the Planck constant, the elementary charge, and the speed of light. In doing so, it anchors our system of measurement in quantities that are directly tied to observation and experiment.

What is striking, however, is what it does not include.

There is no place in the SI framework for the various additional “charges” or quantum numbers that appear in the Standard Model—no color charge, no flavour, no isospin. These concepts may be essential within the mathematical structure of modern particle physics, but they do not enter the system that defines how we measure physical reality.

This is not a flaw in the SI system—quite the contrary. It is designed to remain independent of theoretical interpretation, and to rely only on quantities that can be operationally defined and reproducibly measured.

But that, in itself, is revealing.

It suggests a distinction between what we can measure directly and what we introduce as part of a theoretical framework. And it raises a question—at least for me—about how closely our most advanced theories are tied to physically meaningful quantities.

None of this diminishes the achievements recognized by a Nobel Prize in Physics or other honours—or the remarkable success of modern theoretical physics more generally. But it does serve as a quiet reminder that predictive success is not the same as final understanding.

If anything, the SI revision reinforces my own inclination to look for interpretations of physics that remain as close as possible to what can be directly measured and understood.

Post-Post-Scriptum on what I would like to write

Since writing this, I’ve taken a small but meaningful step: I uploaded a somewhat older manuscript and a newly written Chapter 2 to ResearchGate, as companion documents to my Radial Genesis paper (thoughts on cosmology).

It is not as a finished book — far from it — but as a snapshot of where my thinking currently stands. If I were to write a full-blown book about this, it would not be a technical monograph, nor a speculative manifesto. It would be something in between: a guided journey. I would try to connect three layers:

the physical intuition (what kind of universe are we actually living in?),
the mathematical structure (how symmetry, geometry, and scaling laws shape that intuition),
and the cosmological narrative (how a finite universe with emergent spacetime could naturally arise).

Most importantly, I would try to bridge particle physics and cosmology — not as separate domains, but as different perspectives on the same underlying structure.

The current documents are fragments of that attempt. For now, I will leave them as they are. Sometimes it is better to pause, let ideas settle, and return later with fresh eyes.

Post-post-post-scriptum

I couldn’t help thinking about this question: if the math in academic physics has become “ugly” or “lost,” then what would a beautiful alternative look like? Of course, ‘beauty’ (for me, at least) is a combination of simplicity and realism, and so that is my ‘RealQM’ world view. So I did a quick paper on ResearchGate on what Sabine Hossenfelder still thinks of as very ‘mysterious’ but which, to me, is easily explained in my ‘RealQM’ framework’:

The “Ghost” Sector (Dark Matter): Two types of electromagnetism (defined by the fundamental asymmetry in Maxwell’s equations modern mainstream physicists completely ignore) share the same spacetime but do not interact otherwise. Because they share the same spacetime, they do interact ‘gravitationally’. Full stop: no further explanation needed.
The Proton Radius: My two-line theoretical calculation gives a proton radius of 0.841 fm. Recent measurements clocked the proton at 0.8406(15) fm. What more confirmation is needed to urge physicists to think of particles as dynamical structures rather than abstract entities with lots of abstract or non-measurable properties?
Needless to say: challenges are still out there, and AI baptizes one of them now officially as The Geometry Challenge or Proton Yarnball Puzzle.

Read this last (?) working paper on ResearchGate here.

End of the Road to Reality?

Pre-scriptum (dated 26 June 2020): This post did not suffer from the DMCA take-down of some material. It is, therefore, still quite readable—even if my views on these matters have evolved quite a bit as part of my realist interpretation of QM. I now think the idea of force-carrying particles (bosons) is quite medieval. Moreover, I think the Higgs particle and other bosons (except for the photon and the neutrino) are just short-lived transients or resonances. Disequilibrium states, in other words. One should not refer to them as particles.

Original post:

Or the end of theoretical physics?

In my previous post, I mentioned the Goliath of science and engineering: the Large Hadron Collider (LHC), built by the European Organization for Nuclear Research (CERN) under the Franco-Swiss border near Geneva. I actually started uploading some pictures, but then I realized I should write a separate post about it. So here we go.

The first image (see below) shows the LHC tunnel, while the other shows (a part of) one of the two large general-purpose particle detectors that are part of this Large Hadron Collider. A detector is the thing that’s used to look at those collisions. This is actually the smallest of the two general-purpose detectors: it’s the so-called CMS detector (the other one is the ATLAS detector), and it’s ‘only’ 21.6 meter long and 15 meter in diameter – and it weighs about 12,500 tons. But so it did detect a Higgs particle – just like the ATLAS detector. [That’s actually not 100% sure but it was sure enough for the Nobel Prize committee – so I guess that should be good enough for us common mortals :-)]

The picture above shows one of these collisions in the CMS detector. It’s not the one with the trace of the Higgs particle though. In fact, I have not found any image that actually shows the Higgs particle: the closest thing to such image are some impressionistic images on the ATLAS site. See http://atlas.ch/news/2013/higgs-into-fermions.html

In case you wonder what’s being scattered here… Well… All kinds of things – but so the original collision is usually between protons (so these are hydrogen ions: H⁺nuclei), although the LHC can produce other nucleon beams as well (collectively referred to as hadrons). These protons have energy levels of 4 TeV (tera-electronVolt: 1 TeV = 1000 GeV = 1 trillion eV = 1×10¹² eV).

Now, let’s think about scale once again. Remember (from that same previous post) that we calculated a wavelength of 0.33 nanometer (1 nm = 1×10^–9 m, so that’s a billionth of a meter) for an electron. Well, this LHC is actually exploring the sub-femtometer (fm) frontier. One femtometer (fm) is 1×10^–15 m so that’s another million times smaller. Yes: so we are talking a millionth of a billionth of a meter. The size of a proton is an estimated 1.7 femtometer indeed and, as you surely know, a proton is a point-like thing occupying a very tiny space, so it’s not like an electron ‘cloud’ swirling around: it’s much smaller. In fact, quarks – three of them make up a proton (or a neutron) – are usually thought of as being just a little bit less than half that size – so that’s about 0.7 fm.

It may also help you to use the value I mentioned for high-energy electrons when I was discussing the LEP (the Large Electron-Positron Collider, which preceded the LHC) – so that was 104.5 GeV – and calculate the associated de Broglie wavelength using E = hf and λ = v/f. The velocity v is close to c and, hence, if we plug everything in, we get a value close to 1.2×10^–15 m indeed, so that’s the femtometer scale indeed. [If you don’t want to calculate anything, then just note we’re going from eV to giga-eV energy levels here, and so our wavelength decreases accordingly: one billion times smaller. Also remember (from the previous posts) that we calculated a wavelength of 0.33×10^–6 m and an associated energy level of 70 eV for a slow-moving electron – i.e. one going at 2200 km per second ‘only’, i.e. less than 1% of the speed of light.] Also note that, at these energy levels, it doesn’t matter whether or not we include the rest mass of the electron: 0.511 MeV is nothing as compared to the GeV realm. In short, we are talking very very tiny stuff here.

But so that’s the LEP scale. I wrote that the LHC is probing things at the sub-femtometer scale. So how much sub-something is that? Well… Quite a lot: the LHC is looking at stuff at a scale that’s more than a thousand times smaller. Indeed, if collision experiments in the giga-electronvolt (GeV) energy range correspond to probing stuff at the femtometer scale, then tera-electronvolt (TeV) energy levels correspond to probing stuff that’s, once again, another thousand times smaller, so we’re looking at distances of less than a thousandth of a millionth of a billionth of a meter. Now, you can try to ‘imagine’ that, but you can’t really.

So what do we actually ‘see’ then? Well… Nothing much one could say: all we can ‘see’ are traces of point-like ‘things’ being scattered, which then disintegrate or just vanish from the scene – as shown in the image above. In fact, as mentioned above, we do not even have such clear-cut ‘trace’ of a Higgs particle: we’ve got a ‘kinda signal’ only. So that’s it? Yes. But then these images are beautiful, aren’t they? If only to remind ourselves that particle physics is about more than just a bunch of formulas. It’s about… Well… The essence of reality: its intrinsic nature so to say. So… Well…

Let me be skeptical. So we know all of that now, don’t we? The so-called Standard Model has been confirmed by experiment. We now know how Nature works, don’t we? We observe light (or, to be precise, radiation: most notably that cosmic background radiation that reaches us from everywhere) that originated nearly 14 billion years ago (to be precise: 380,000 years after the Big Bang – but what’s 380,000 years on this scale?) and so we can ‘see’ things that are 14 billion light-years away. In fact, things that were 14 billion light-years away: indeed, because of the expansion of the universe, they are further away now and so that’s why the so-called observable universe is actually larger. So we can ‘see’ everything we need to ‘see’ at the cosmic distance scale and now we can also ‘see’ all of the particles that make up matter, i.e. quarks and electrons mainly (we also have some other so-called leptons, like neutrinos and muons), and also all of the particles that make up anti-matter of course (i.e. antiquarks, positrons etcetera). As importantly – or even more – we can also ‘see’ all of the ‘particles’ carrying the forces governing the interactions between the ‘matter particles’ – which are collectively referred to as fermions, as opposed to the ‘force carrying’ particles, which are collectively referred to as bosons (see my previous post on Bose and Fermi). Let me quickly list them – just to make sure we’re on the same page:

Photons for the electromagnetic force.
Gluons for the so-called strong force, which explains why positively charged protons ‘stick’ together in nuclei – in spite of their electric charge, which should push them away from each other. [You might think it’s the neutrons that ‘glue’ them together but so, no, it’s the gluons.]
W⁺, W^–, and Z bosons for the so-called ‘weak’ interactions (aka as Fermi’s interaction), which explain how one type of quark can change into another, thereby explaining phenomena such as beta decay. [For example, carbon-14 will – through beta decay – spontaneously decay into nitrogen-14. Indeed, carbon-12 is the stable isotope, while carbon-14 has a life-time of 5,730 ± 40 years ‘only’ 🙂 and, hence, measuring how much carbon-14 is left in some organic substance allows us to date it (that’s what (radio)carbon-dating is about). As for the name, a beta particle can refer to an electron or a positron, so we can have β^– decay (e.g. the above-mentioned carbon-14 decay) as well as β⁺decay (e.g. magnesium-23 into sodium-23). There’s also alpha and gamma decay but that involves different things. In any case… Let me end this digression within the digression.]
Finally, the existence of the Higgs particle – and, hence, of the associated Higgs field – has been predicted since 1964 already, but so it was only experimentally confirmed (i.e. we saw it, in the LHC) last year, so Peter Higgs – and a few others of course – got their well-deserved Nobel prize only 50 years later. The Higgs field gives fermions, and also the W⁺, W^–, and Z bosons, mass (but not photons and gluons, and so that’s why the weak force has such short range – as compared to the electromagnetic and strong forces).

So there we are. We know it all. Sort of. Of course, there are many questions left – so it is said. For example, the Higgs particle does actually not explain the gravitational force, so it’s not the (theoretical) graviton, and so we do not have a quantum field theory for the gravitational force. [Just Google it and you’ll see why: there’s theoretical as well as practical (experimental) reasons for that.] Secondly, while we do have a quantum field theory for all of the forces (or ‘interactions’ as physicists prefer to call them), there are a lot of constants in them (much more than just that Planck constant I introduced in my posts!) that seem to be ‘unrelated and arbitrary.’ I am obviously just quoting Wikipedia here – but it’s true.

Just look at it: three ‘generations’ of matter with various strange properties, four force fields (and some ‘gauge theory’ to provide some uniformity), bosons that have mass (the W⁺, W^–, and Z bosons, and then the Higgs particle itself) but then photons and gluons don’t… It just doesn’t look good, and then Feynman himself wrote, just a few years before his death (QED, 1985, p. 128), that the math behind calculating some of these constants (the coupling constant j for instance, or the rest mass n of an electron), which he actually invented (it makes use of a mathematical approximation method called perturbation theory) and for which he got a Nobel Prize, is a “dippy process” and that “having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent“. He adds: “It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization [“the shell game that we play to find n and j” as he calls it] is not mathematically legitimate.” And so he writes this about quantum electrodynamics, not about “the rest of physics” (and so that’s quantum chromodynamics (QCD) – the theory of the strong interactions – and quantum flavordynamics (QFD) – the theory of weak interactions) which, he adds, “has not been checked anywhere near as well as electrodynamics.”

Waw ! That’s a pretty damning statement, isn’t it? In short, all of the celebrations around the experimental confirmation of the Higgs particle cannot hide the fact that it all looks a bit messy. There are other questions as well – most of which I don’t understand so I won’t mention them. To make a long story short, physicists and mathematicians alike seem to think there must be some ‘more fundamental’ theory behind. But – Hey! – you can’t have it all, can you? And, of course, all these theoretical physicists and mathematicians out there do need to justify their academic budget, don’t they? And so all that talk about a Grand Unification Theory (GUT) is probably just what is it: talk. Isn’t it? Maybe.

The key question is probably easy to formulate: what’s beyond this scale of a thousandth of a proton diameter (0.001×10^–15m) – a thousandth of a millionth of a billionth of a meter that is. Well… Let’s first note that this so-called ‘beyond’ is a ‘universe’ which mankind (or let’s just say ‘we’) will never see. Never ever. Why? Because there is no way to go substantially beyond the 4 TeV energy levels that were reached last year – at great cost – in the world’s largest particle collider (the LHC). Indeed, the LHC is widely regarded not only as “the most complex and ambitious scientiﬁc project ever accomplished by humanity” (I am quoting a CERN scientist here) but – with a cost of more than 7.5 billion Euro – also as one of the most expensive ones. Indeed, taking into account inflation and all that, it was like the Manhattan project indeed (although scientists loathe that comparison). So we should not have any illusions: there will be no new super-duper LHC any time soon, and surely not during our lifetime: the current LHC is the super-duper thing!

Indeed, when I write ‘substantially‘ above, I really mean substantially. Just to put things in perspective: the LHC is currently being upgraded to produce 7 TeV beams (it was shut down for this upgrade, and it should come back on stream in 2015). That sounds like an awful lot (from 4 to 7 is +75%), and it is: it amounts to packing the kinetic energy of seven flying mosquitos (instead of four previously :-)) into each and every particle that makes up the beam. But that’s not substantial, in the sense that it is very much below the so-called GUT energy scale, which is the energy level above which, it is believed (by all those GUT theorists at least), the electromagnetic force, the weak force and the strong force will all be part and parcel of one and the same unified force. Don’t ask me why (I’ll know when I finished reading Penrose, I hope) but that’s what it is (if I should believe what I am reading currently that is). In any case, the thing to remember is that the GUT energy levels are in the 10¹⁶ GeV range, so that’s – sorry for all these numbers – a trillion TeV. That amounts to pumping more than 160,000 Joule in each of those tiny point-like particles that make up our beam. So… No. Don’t even try to dream about it. It won’t happen. That’s science fiction – with the emphasis on fiction. [Also don’t dream about a trillion flying mosquitos packed into one proton-sized super-mosquito either. :-)]

So what?

Well… I don’t know. Physicists refer to the zone beyond the above-mentioned scale (so things smaller than 0.001×10^–15m) as the Great Desert. That’s a very appropriate name I think – for more than one reason. And so it’s this ‘desert’ that Roger Penrose is actually trying to explore in his ‘Road to Reality’. As for me, well… I must admit I have great trouble following Penrose on this road. I’ve actually started to doubt that Penrose’s Road leads to Reality. Maybe it takes us away from it. Huh? Well… I mean… Perhaps the road just stops at that 0.001×10^–15m frontier?

In fact, that’s a view which one of the early physicists specialized in high-energy physics, Raoul Gatto, referred to as the zeroth scenario. I am actually not quoting Gatto here, but another theoretical physicist: Gerard ‘t Hooft, another Nobel prize winner (you may know him better because he’s a rather fervent Mars One supporter, but so here I am referring to his popular 1996 book In Search of the Ultimate Building Blocks). In any case, Gatto, and most other physicists, including ‘T Hooft (despite the fact ‘T Hooft got his Nobel prize for his contribution to gauge theory – which, together with Feynman’s application of perturbation theory to QED, is actually the backbone of the Standard Model) firmly reject this zeroth scenario. ‘T Hooft himself thinks superstring theory (i.e. supersymmetric string theory – which has now been folded into M-theory or – back to the original term – just string theory – the terminology is quite confusing) holds the key to exploring this desert.

But who knows? In fact, we can’t – because of the above-mentioned practical problem of experimental confirmation. So I am likely to stay on this side of the frontier for quite a while – if only because there’s still so much to see here and, of course, also because I am just at the beginning of this road. 🙂 And then I also realize I’ll need to understand gauge theory and all that to continue on this road – which is likely to take me another six months or so (if not more) and then, only then, I might try to look at those little strings, even if we’ll never see them because… Well… Their theoretical diameter is the so-called Planck length. So what? Well… That’s equal to 1.6×10⁻³⁵m. So what? Well… Nothing. It’s just that 1.6×10⁻³⁵m is 1/10 000 000 000 000 000 of that sub-femtometer scale. I don’t even want to write this in trillionths of trillionths of trillionths etcetera because I feel that’s just not making any sense. And perhaps it doesn’t. One thing is for sure: that ‘desert’ that GUT theorists want us to cross is not just ‘Great’: it’s ENORMOUS!

Richard Feynman – another Nobel Prize scientist whom I obviously respect a lot – surely thought trying to cross a desert like that amounts to certain death. Indeed, he’s supposed to have said the following about string theorists, about a year or two before he died (way too young): “I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation–a fix-up to say, “Well, it might be true.” For example, the theory requires ten dimensions. Well, maybe there’s a way of wrapping up six of the dimensions. Yes, that’s all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there’s no reason whatsoever in superstring theory that it isn’t eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn’t produce anything; it has to be excused most of the time. It doesn’t look right.”

Hmm… Feynman and ‘T Hooft… Two giants in science. Two Nobel Prize winners – and for stuff that truly revolutionized physics. The amazing thing is that those two giants – who are clearly at loggerheads on this one – actually worked closely together on a number of other topics – most notably on the so-called Feynman-‘T Hooft gauge, which – as far as I understand – is the one that is most widely used in quantum field calculations. But I’ll leave it at that here – and I’ll just make a mental note of the terminology here. The Great Desert… Probably an appropriate term. ‘T Hooft says that most physicists think that desert is full of tiny flowers. I am not so sure – but then I am not half as smart as ‘T Hooft. Much less actually. So I’ll just see where the road I am currently following leads me. With Feynman’s warning in mind, I should probably expect the road condition to deteriorate quickly.

Post scriptum: You will not be surprised to hear that there’s a word for 1×10^–18m: it’s called an attometer (with two t’s, and abbreviated as am). And beyond that we have zeptometer (1 zm = 1×10^–21m) and yoctometer (1 ym = 1×10^–23m). In fact, these measures actually represent something: 20 yoctometer is the estimated radius of a 1 MeV neutrino – or, to be precise, its the radius of the cross section, which is “the effective area that governs the probability of some scattering or absorption event.” But so then there are no words anymore. The next measure is the Planck length: 1.62 × 10⁻³⁵ m – but so that’s a trillion (10¹²) times smaller than a yoctometer. Unimaginable, isn’t it? Literally.

Note: A 1 MeV neutrino? Well… Yes. The estimated rest mass of an (electron) neutrino is tiny: at least 50,000 times smaller than the mass of the electron and, therefore, neutrinos are often assumed to be massless, for all practical purposes that is. However, just like the massless photon, they can carry high energy. High-energy gamma ray photons, for example, are also associated with MeV energy levels. Neutrinos are one of the many particles produced in high-energy particle collisions in particle accelerators, but they are present everywhere: they’re produced by stars (which, as you know, are nuclear fusion reactors). In fact, most neutrinos passing through Earth are produced by our Sun. The largest neutrino detector on Earth is called IceCube. It sits on the South Pole – or under it, as it’s suspended under the Antarctic ice, and it regularly captures high-energy neutrinos in the range of 1 to 10 TeV. Last year (in November 2013), it captured two with energy levels around 1000 TeV – so that’s the peta-electronvolt level (1 PeV = 1×10¹⁵eV). If you think that’s amazing, it is. But also remember that 1 eV is 1.6×10⁻¹⁹Joule, so it’s ‘only’ a ten-thousandth of a Joule. In other words, you would need at least ten thousand of them to briefly light up an LED. The PeV pair was dubbed Bert and Ernie and the illustration below (from IceCube’s website) conveys how the detectors sort of lit up when they passed. It was obviously a pretty clear ‘signal’ – but so the illustration also makes it clear that we don’t really ‘see’ at such small scale: we just know ‘something’ happened.