Earlier this summer, I decided to read up on Category Theory and Topos.
As you know, we have been writing research articles on Gate model and on Adiabatic Quantum Computing.
These are the two dominant computational models in quantum computing. But they are not the only ones.
Another one is Measurement Based Quantum Computing (MBQC), on something called “one-way quantum computer.”
We want to study MBQC. Topos seemed like the appropriate mathematical formalism to use as it seems to naturally handle contextuality and non-Boolean (Intuitionist) logic.
I like pure mathematics for its own sake, and so that is one reason for me to want to learn something new like Topos.
Even before MBQC motivation, I wanted to know more about intuitionist logic, especially the “law of excluded middle”. It has many implications, one of them being that “proof by contradiction” may not be valid as demonstrating the truth of a proposition and so only constructive methods are considered as (legitimate) proof.
More pragmatically, pure mathematics has been a very rewarding plunder-box – a veritable treasure chest — to understand our physical Universe. Just ask Einstein.
There is this braided relationship between physics and mathematics. It has been there for centuries, if not for millennia.
Mathematics has served physics well. In return, physics has gifted mathematics with some intricate challenges that have led to beautiful mathematics.
I have always thought about the intertwining of mathematics and physics in a certain way, as a virtuous cycle that feed off positively on each other.
Having said that, I recently re-read Physics and Philosophy by James Jeans:
The history of theoretical physics is a record of the clothing of mathematical formulae which were right, or very nearly right, with physical interpretations which were often very badly wrong.
Newton put science on the wrong track for two centuries by interpreting his laws of motion in terms of forces and absolute space and time.
Not just Newton, but also Maxwell. We now know that:
…electric and magnetic forces are not real at all; they are mere mental constructs of our own, resulting from our rather misguided efforts to understand the motions of particles.
Then, I read somewhere that mathematical physicist is not the same as a theoretical physicist.
What is the difference, you may also wonder?
A theoretical physicist is a physicist who uses mathematics as needed to crisply understand physical phenomenon.
A mathematical physicist is a mathematician who finds the physicist’s use of mathematics too clumsy and irritating, and so proceeds to provide secure and rigorous formalism to the allegedly careless mathematical constructions of physicists.
I had so naively thought of mathematicians and physicists as best-friends, or as a couple in a happy marriage. But evidently, they had a bitter divorce many decades back, and if anything, have been making some small efforts recently to reunite.
I hope they do reunite. When they dance well, they dance really well.
As I like General Relativity (GR), I wondered if the pictorial description typically presented to a layperson of “heavy ball curving a rubber sheet” was all there is to it. Of course not. It is way, way deeper.
Of course, one can (and supposedly should, if one really wants to do GR) dispose of the pictorial description, and work directly with the mathematics.
But this is not without its problems, as I now understand. Even setting aside quantum mechanics, the underlying mathematics of GR uses points and real numbers, and smooth manifolds and Riemann geometry, and so on, and from what I can gather this is the problem (as it is with any field theory that relies on points), not towards making very accurate predictions (clearly, as GR certainly does make very accurate predictions), but for it to be regarded as a fundamental theory.
That is, GR is just a very nice approximation of something far more intricate (that we do not know yet), that may or may not have any resemblance to the pictorial “heavy ball curving a rubber sheet” interpretation of GR, and, importantly, uses mathematical machinery that is very different from the one in use today.
What could that very different math be? You guessed it: Topos.😏
Quantum Mechanics (QM) has always had its detractors, many of whom are considered as founders (Planck and Einstein among them) of QM!
From 1900 to late 1920s, QM was built as a patchwork of ad-hoc fragments, with not-so-polite back-and-forth between Schrodinger and Heisenberg, while magnificently explaining many baffling puzzles that had defied classical physics, such as black-body radiation and photoelectric effect.
In 1932, John von Neumann gave QM a stable foundation, that of Hilbert Space formalism, which continues to be the mainstream workhorse.
But this is not without its problems either. Again, it is not just in the interpretation (Copenhagen, many-worlds etc), but in its incompatibility with an essential basis – indeed a sine qua non – of GR, which indissolubly unites space and time.
Of course, the dream of unifying GR and QM has been there since the 1920s.
Before you can unite GR and QM, it seemed sensible to first try to unite Special Relativity (SR) and QM.
As George Polya said:
If there is a hard problem you cannot solve, there is an easier problem that you cannot solve. Find it. (And solve that first.)
Progress in finding ways to force-fit SR and QM have led to spectacular outcomes, such as the prediction (and then actually finding) of anti-particles.
More has been accomplished in the four decades following the work of Dirac, such as Quantum Electro-Dynamics (QED) and Quantum Chromo-Dynamics (QCD), which form important constituents of the Standard Model.
But the unification of GR with QM has not been accomplished.
(Furthermore, many physicists consider the Standard Model as way too ugly, although extra-ordinarily accurate, and so believe something more fundamental exists that will be satisfyingly beautiful.)
It is not for the lack of trying though. Thousands of researchers, for decades, have been working on this with great vigor.
They seem to be getting frustrated. And desperate. They are thrashing around with all sorts of “imaginative ideas” un-tempered by experiments, or even, it appears now, with experiment-ability.
Strings, 26 dimensions (or is it 10?), supersymmetry….oh, come on, folks.
Claims of the imminent unification of GR with QM, however, have been like crying wolf amplified by a hysterical news-media (and physicist authored books for the layperson) covering “beauty” and “scientific advances.”
Fake news? Fake science!
Recently, I read somewhere (maybe in Sabine Hossenfelder’s cri de coeur, Lost in Math) that some physicists are now asking that the rules of science be changed to not have experimental verification as a criterion for accepting their (“so beautiful it has to be this way”) theories!
That society should rely on the collective confabulation (a phrase I liked from Tim Lewens’ The Meaning of Science) of these physicists based on group think.
Fake scientists? These theoretical physicists make organized religion look hyper-rational!
Oh, Galileo, how you must be spinning in your grave.
What about mathematical physicists? Do they have something to offer towards the unification of GR and QM?
Yes, evidently. One of the suggestions is to jettison the Hilbert Space formalism, in its entirety (yes, in toto), and replace it with, you guessed it, Topos. 😊
I do know a couple of theoretical physicists, and so I thought I would ask them what they thought about Topoi, Heyting algebras, functors, locales, pointless presheaves, truth-values and such.
This reminds me of “truthiness” coined by Stephen Colbert!
Prabha Mandayam (PhD Caltech, in physics, with John Preskill as advisor, and so a bona fide theoretical physicist), a faculty member in the physics department at my alma-mater IIT-Madras, calmly responded that “it is unlikely that we will abandon Hilbert Basis formalism anytime soon” and then said something that startled me:
Remember last time we were discussing mathematics in ancient India? You might want to know that they did not consider proof by contradiction as a valid argument. They insisted on a constructive proof.
Aryabhatta, Bhaskara I, Sridhara, Bhaskara II, Madhava……channeling Calvin Candle (played by Leonardo DiCaprio) from Quentin Tarantino’s Django Unchained:
Gentlemen, you had my curiosity. But now you have my attention.
If we are going to have fun with mental constructs and imaginative ideas, why not map to Marvel Universe😊:
GR: QM = Thor: Loki?
I wonder if Einstein and Von Neumann, like Newton and Maxwell, unwittingly, have set back science by a century?
Transferring Ronald Reagan’s insight on government to this context:
Einstein and von Neumann are not the solution. Einstien and von Neumann are the problem.
Who better to ask about quantum foundations than CMU’s own Bob Griffiths?
So I emailed him. His response:
I do not know the book. Halvorson is a philosopher who was at one time in Pittsburgh; may have completed his doctorate here.
Yes, I checked. He is a Professor of Philosophy at Princeton.
A gem of a book I recall reading, written by Harry G. Frankfurt, also a Princeton Professor of Philosophy, is “On Bullshit” (you may have seen this discussed on The Daily Show with Jon Stewart):
One of the most salient features of our culture is that there is so much bullshit.
..not even concerned whether her statement is correct. Her fault is not that she fails to get things right, but that she is not even trying. She is not concerned with the truth-value of what she says.
It is impossible for someone to lie unless he thinks he knows the truth. Producing bullshit requires no such conviction. He does not care whether the things he says describe reality correctly. He just picks them out, or makes them up, to suit his purpose.
By virtue of this, bullshit is a greater enemy of the truth than lies are.
Perhaps, I should use my RAGS Family Foundation to create a “No-Bull” Prize in Physics!
Will there be any candidates for consideration, though?
I think this was in King Lear:
King: I can hail the spirits from the vasty deep.
Slave: So can I, my lord, as can any other man. But, will they come when you call for them?
I am not a physicist. I am not a mathematician. I am not a philosopher.
I am just an Academic Capitalist, seeking intellectual enjoyment through pure mathematics that could lead to decent commercial payoff when quantum computers become available.
A recent National Academies consensus report Quantum Computing: Progress and Prospects says that I should not hold my breath.
Just intellectual enjoyment. For now.
Fascinating. Thanks for sharing, even though most of the content is way over my head. (Special Relativity is as far as I got. GR might as well be Sanskrit. String Theory cannot possibly be anything but angels-dancing-on-the-head-of-a-pin bullshit.)
Here is my layman’s reaction.
There is not a human endeavor that Government cannot fuck up by showering it with money. Whether it’s healthcare or real estate or banking or education, if rivers of government money flow into a field for a long enough period of time it will get fucked up.
I’m sure you are following the reproducibility crisis that is plaguing Big Science. I follow it very closely in the life science space. You’re probably also following the predatory journal crisis. I won’t even mention the AGW hysteria crisis as those very words induce religious fits in true believers. But, Sridhar, you cannot doubt that Science is all fucked up.
It’s time to shut off the river of taxpayer money that Vannevar Bush first directed into Big Science. At the very least the herd needs to be thinned. There is no longer any reason to keep armies of second and third rate scientists on the public dole, even if there was a good reason in the first place. Let a far smaller number of exceptional first rate scientists truly dedicated to their work and not just there to enjoy the lifestyle sort it out.
My two cents. As a taxpayer, science lover, and former engineer.
Category Theory..Hmm. Long time ago, recall reading about functors! Thats all I remember now!!
Interesting write up. Enjoyed reading it.