The Buddha, Topoi, And Quantum Gravity
By Amir D. Aczel | August 24th 2012 10:28 PM | 45 comments | Print | E-mail | Track Comments

Amir D. Aczel, Ph.D., studied mathematics and physics at the University of California at Berkeley and also holds a doctorate in statistics. He is...

View Amir D.'s Profile
Anything is either true,
Or not true,
Or both true and not true,
Or neither true nor not true;
This is the Buddha's teaching.

--Nagarjuna (second century Buddhist monk and philosopher), the Mulamadhyamakakarika, Chapter XVIII, verse 8 (Note: there are other translations of this verse, for example, using "real" instead of "true.")

To a Western mind, this may seem like complete nonsense. True and not true are mutually exclusive and exhaustive states for any proposition--in fact, the law of the excluded middle, as this idea is expressed in mathematics, is the basis for much of (a certain kind of) proof theory. If you disallow it, proof by contradiction would not hold, and many theorems in mathematics would be unproved and undecided. So, what's behind this puzzling statement attributed to the Buddha?--and why should we care?

I recently came across a fascinating article by the mathematician Fred Linton of Wesleyan University, which explains the Buddhist idea of the four possibilities in the verse above in a logical, mathematical way. Before getting to the math, let's look at some everyday examples that Linton provides for situations where the additional two logical possibilities may hold: both true and not true; and neither true nor not true. If you have a student, Linton writes, who is brilliant in mathematics but also has a knack for getting arrested in campus demonstrations, you might rightly say that he is both very bright and not very bright. A cup of coffee with just a small amount of sugar, Linton points out, could very well be described as neither sweet nor not sweet. Such examples abound.

Apparently, Eastern thinking is more in line with such gradations of truth and falsity, so that the law of the excluded middle doesn't quite apply. In a sense, the strict interpretation that anything must be either true or not true may well represent a Western bias in thinking about nature and life. In an e-mail message, Linton provided more examples, of the other kind, and which are obviously Western in their strict "either or" bias: "You are either with me or against me"; "If you're not part of the solution, you're part of the problem"; and: "Which will you have?--tea or coffee?" Then Linton added: "I blame Aristotle for that!"

In fact, our Western logic does go back to Aristotle, famous for logical deductive statements such as: "All men are mortal; Socrates is a man. Therefore, Socrates is mortal." Sure--but there are other kinds of logic as well, and they apply in various situations. Eastern thinking modes seem to be more accepting of these differing ways of thinking. But the question arises: Isn't it true that mathematics brings us naturally to the Western "either or" kind of logic? Surprisingly, the real answer is: Not necessarily!

From Greece of the time of Aristotle (the fourth century B.C.), let's jump to the middle of the twentieth century, and find ourselves in Paris. Alexander Grothendieck (born in 1928), one of the brightest and yet most troubled mathematicians of all time (he is currently living in hiding in the French Pyrenees, having completely withdrawn from society believing that the Devil rules the world--See Pierre Cartier, "A Mad Day's Work," Bulletin of the American Mathematical Society 38, No. 4, 2001, p. 393) is recasting the entire field of algebraic geometry, and as part of that brilliant undertaking he invents a new concept: the topos. A topos means "location," or "place"--the plural, since this is a Greek word, is topoi; but in English we often also call them "toposes"--and it is the ultimate generalization of the concept of space

Only Grothendieck could have the audacity, and the incredible facility with mathematics, to dare propose such a bold idea. Then, according to Cartier, "Grothendieck claimed the right to transcribe mathematics into any topos whatever." (Cartier, op. cit., p. 395; italics in the original.) Grothendieck, followed by William Lawvere and Miles Tierney, used category theory (an extremely general and abstract branch of mathematics championed by Saunders MacLane and Samuel Eilenberg) to recast mathematics into a new and far more supple mold. What did they do?

The problem with our Western, Aristotelian logic--which leaves no room between "true" and "not true"--is its crucial reliance on set theory. In set theory, which is often seen as the foundation of mathematics (surely you've heard everything about ZFC--Zermelo and Fraenkel's system; Cantor, Russell, and Gödel), a point is either a member of a set, or not a member of the set: there is nothing in between. This set-theoretic logic leads to the familiar Boolean algebra, which rules the truth or falsity of everything: unions, intersections, negations, and implications of events--and it forms the basis for computer science.

But with its powers, also come its limitations: Set theory cannot handle the in-between, the no-man's-land between truth and non-truth, being and non-being: the natural fuzziness in so many of life's situations. What Grothendieck's work brought to the fore was that there are other kinds of mathematics. Set theory is not the only possible foundation for mathematics: Topoi provide other--equally valid, if not more so (because they may avoid the paradoxes of set theory)--foundations. The key idea is that in a topos, which is based on the very general theory of categories, there is no necessity for set membership. In category theory, instead of sets, and set membership (which is too restrictive), we have objects and mappings (often called simply "arrows") between objects. The objects are very general, and so are the mappings. We write: f: a --> b; an object a is mapped by the arrow (the mapping) f into the object b. The generality of this approach is stunning at first, but mathematicians get used to it. What this device achieves is that it replaces the set-membership idea with the arrow idea. You no longer view things internally (using set membership), but rather externally: by how something is mapped, or transformed, into something else. The theory is very deep, and extremely powerful. Category theory relies on diagrams of objects and arrows, and a key concept that teaches us about the underlying mathematics is whether a diagram commutes or not (is going from A to C directly the same as going from A to B and then to C?--if the diagram commutes, you get a mathematical identity: f = gh, for the corresponding mappings, the product identified as map composition; and this can be extended).

The topos was born within the context of category theory, but it has an additional, key property. A topos also has a topological structure. In topology, we study spaces and continuous functions. One important concept in topology is the idea of nearness. It turns out that this idea of a distance between a point and a set in topology can very nicely replace the idea of set membership (it's important to note, however, that the category of sets is also a topos--but there are others, and they provide the freedom we need from set membership).

So instead of the "true or false" structure derived from set theory, we can construct topoi in which this idea is mathematically replaced by a particular topological distance function, which allows for situations such as "both true and false" or "neither true nor false." It is, in fact, such a topos that Fred Linton exhibits in his paper mentioned above. Other topoi, with other kinds of generality and structure, are also possible. And all of them are mathematically valid! Before we move to physics, I want to recommend some books on category theory and topos theory. For categories, Saunders MacLane's book, Categories for the Working Mathematician, Springer, 1998, is the classic reference. For topoi, I highly recommend the very readable Topoi: The Categorial Analysis of Logic, by Robert Goldblatt, Dover, 1979.

So, unless you've never heard the expression "quantum weirdness"--in which case you must have come to this site by an unlikely accident of fate--you know by now why I am so enamored with topoi. But let's start with the simplest quantum idea: the Young double-slit experiment. You shoot a single photon at a barrier with two slits. What can the photon do? Go through slit 1, or go through slit 2, or go through both slit 1 and slit 2 (of course these depend on conditions we won't get into--but we all know this story; Richard Feynman called this "the only quantum mystery" since it embodies all the key elements). So you say that this covers the first three lines of the Buddha's choices according to Nagarjuna (in reference to one of the two slits): True, False, both True and False. But what about the fourth one: Neither True nor False? Well, so the Buddha didn't know everything--but at least he did get us away from our "it must be either true or false" hangup! But...hold it! If the photon has a high enough energy, it can turn into an electron-positron virtual pair. In this case, it would go through neither slit 1 nor slit 2. That Buddha was pretty smart after all!

So you'd think that topos theory should be pretty promising as a new foundation for quantum mechanics. Indeed, it is! And the great value that topoi add here is that they allow us to get away from the--artificial, in my view--reliance on probabilities (the Copenhagen interpretation), or even more outlandish interpretations of quantum weirdness: Many Worlds, or the de Broglie-Bohm approach, or anything else. We may find a topos that describes quantum logic directly from the mathematics (where the new topos--and Linton's is a good candidate--replaces the usual mathematical universe). We now have no need to resort to "collapsing the wave function" and then relying on the mysteriously appearing "probabilities," or on any of the other devices theoretical physicists use to describe quantum behavior.

It turns out that in 1997, the idea was discovered by Chris Isham of Imperial College, London, who wrote a key article about using topos theory in quantum mechanics (you can find it on arXiv). By 2011, several articles providing more detail about how topos theory could be used to explain quantum mechanics in a natural, mathematically-powerful way were published by Isham and his colleague Andreas Döring at Imperial College (all are on arXiv), so that a virtual revolution is now taking place in the foundations of physics. Why haven't you heard about it? The answer is probably: fashion. String theory, despite its setbacks, is still very sexy. People close to the media, such as Brian Greene, toot its horn--so the public hears much about it. And the recent Milner Fundamental Physics Prize (\$3 million!) was awarded to several string theorists.

But recent work, by Chris Isham, C. Flori of the Perimeter Institute, and others, has begun to apply topos theory to quantum gravity--the realm in which string theory was supposed to do its magic--and this provides science with yet another outlet through which to pursue the Holy Grail of wedding quantum mechanics to Einstein's general theory of relativity. Because of the great power, generality, and versatility of topos theory, I vote for this mathematical theory as a likely candidate for producing the coveted "Theory of Everything."

outlandish interpretations of quantum weirdness: Many Worlds, ... relying on the mysteriously appearing "probabilities," ... Why haven't you heard about it? The answer is probably: fashion. String theory, despite its setbacks, is still very sexy.
String theory, although it is a fashion and detrimental to an important part of physics, is not a fashion among people who work on the fundaments of QM, who mostly have no opinion on strings one way or other.
I have a better explanation: We do not care about whatever is inconsistent with "many world" approaches (in the widest sense), simply because such is incompatible with QM. Many in the fundamental QM community try to keep avoiding "many worlds" because it is so unfashionable with the older guard, resulting in desperate formulations of many worlds without "many worlds". I suggest to rethink that strategy. Younger guys do not look at anything that advertises itself to be "not many world nonsense" because such is pseudoscience just like "not relativistic nonsense", no matter whether there are some mathematical gems hidden.

In his most recent blog, Lubos Motl tears apart any 'Many World' interpretation. I don't always agrees with him (and certainly not with his arrogant attitude) but his comments on this subject seem to make sense. BTW From your picture you don't seem much younger than Lubos...

Hi Anonymous, No--I'm probably older than Lubos (judging by his picture...). I don't like "Many Worlds" either, and in my debate with Brian Greene on C-Span I did try to argue against it--but I didn't come out very well; Brian was very argumentative (he was promoting his book on the multiverse, which of course relies on "many worlds" as one of the major chapters...) :)
Amir D. Aczel
Lubos Motl is the guy to go to if you need help with some pure mathematics. Anything else, he does not grasp. It is somewhat of an idiot savant syndrom with him. One tolerates his nonsense against women and alternatives to string theory because at times he has something insightful to say about supersymmetry transformations or quaternions. All else, like what "many worlds" even means, he sadly does not get, because it requires the partially social skill of a sympathetic reading of other's writings. (And yes, he is probably my age - we climbed a mountain once together in Colorado, discussing all kinds of things. A real lesson in psychology that I will never forget. There will always be a place for Lubos in my heart, however much he rants against me.)
You're right, Sascha! I got ahead of myself...the comparison with string theory was supposed to come after the mention of quantum gravity. On the other hand, topos-thinking is not something that theoretical physicists are used to doing, and maybe this is an area of mathematics where methods and results need to be developed further by mathematicians before it can be fruitful as a powerful tool in physics. Perhaps we need another Ed Witten to develop the math for physics. Incidentally: part of the reason Grothendieck (the natural candidate for doing so) has disappeared into the Pyrenees is that he found out that the French Department of Defense was supporting "his" institute--the IHES, outside Paris--and he views all physics as evil because of the connection with the atom bomb (before his disappearance he was very active in antiwar, antinuclear, and environmental causes and also founded a movement called Survivre Pour Vivre).
Amir D. Aczel
OK - you are of course correct to say that if it comes to QM-gravity, strings being too fashionable is an issue (in many ways).
My main problem (overshadowing disagreements on Buddhism and so on) is your (and Isham's etc) unwillingness to let go of naive realisms. Modal realism (~ many worlds) is philosophically self-evident and any formulation (topoi or squiggledypops) is either going to be consistent with it or nonsense. The recent wave of many worlds without mentioning "many worlds" is a waste of resources. Lets just grow up already - Everett relativity is as much cultural relativism and nonsense as Einstein relativity. Just accept it and you will see that nobody really cares, nobody is going to start eating children because of too much relativity and constructionism in science.
Very nice article, thanks for highlighting the tetralemma article! However, one minor quibble: the book Toposes and Local Set Theory is by the mathematician John Lane Bell, not by John Stewart Bell, him of The Theorem...

Thanks!! This is embarrassing! I'll correct it now.
Amir D. Aczel
Thanks for the nice introduction to topoi and how they might be applied in QM, very nicely written!

I do have a question though, I'm no physician or mathematician so bear with me;
How could topoi replace the QM intepretation? Wouldn't using topoi only provide means to make things workout nicely on paper? I thought physics is the act of translating physical observations to a model using mathematics, but there's always the physical side which we need to understand. How would you interpret the concept of such a topos to the physical?

Good question! I think the answer is in the philosophy: we tend to think in a yes-no kind of strictness; a topos simply provides a consistent mathematical universe in which in-between states make logical sense. I don't think that the actual computations of QM would change--but I don't know. In terms of quantum gravity, perhaps the new setting of another topos can offer a natural unification of the two areas, QM and GR. Let's not forget that, in fact, string theory also offers a milieu in which methods from algebra (groups and other structures) are combined with a topological idea. The best example from string theory is of course the Calabi-Yau manifold:

Amir D. Aczel
Good question! I think the answer is in the philosophy: we tend to think in a yes-no kind of strictness

But in defense of the boolean way of thinking, the examples given seem to make statements about complex/compound systems, which could be broken up into more atomic parts where boolean like statements would make sense;

If you have a student, Linton writes, who is brilliant in mathematics but also has a knack for getting arrested in campus demonstrations, you might rightly say that he is both very bright and not very bright

In this example "bright" covers both mathematical and social intelligence. Boolean assignment would work just fine if one would not try to judge his overal intelligence but each aspect individually.

A cup of coffee with just a small amount of sugar, Linton points out, could very well be described as neither sweet nor not sweet

It could be attributed to another state distinct from sweet; "neutral". If all other possible states are nested in available primary choices (sweet, bitter..), neutral could be an empty disjoint set, resulting in neither sweet or not sweet.

Would such topoi / fuzzy logic be more of a convience tool to work with complex systems then? A way of making statements about complex systems Which cannot easily be broken up into more atomic parts?

As you probably (0.9) are aware, there is the entire field of fuzzy logic.  True = 1, False = 0, and Fuzzy is everywhere inbetween.  This allows one to turn statements from impossible puzzles to simple algebra problems that have solutions.  Consider a card that says on one side:
"the statement on the other side is false."

Flip the card, and it says:

"the statement on the other side is true."

Some have written this shows a paradox.  Let's convert this to an algebra problem.  We don't know the value of truth on either side of the card, so make it a variable, X.  Convert the statements to algebra:

"the statement on the other side is false."

1 - X

"the statement on the other side is true."
X

Is there a value of X that makes sense?

1 - X = X
iff
X = 0.5

So both statements are half truths.

My own slant of the slit experiment is that people omit that the source must be coherent in time, or space, or spacetime.  When I tried to make the case for this technical point, quite a few commenters did not accept it.  One can use an incoherent light source such as a flashlight so long as it first goes through a single slit before going through the two slits.  The single slit makes the source coherent in space.  A flashlight shown on the two slits without the pre-single-slit will not show an interference pattern.

Were I to debate the issue again, I would ask my opponent to write down the way they calculate the interference pattern seen on the screen.  They would write down something using an exponential or sines and cosines.  Bingo, bingo, those trig functions are coherent, as in incredibly well organized, ie not random.

The organization of the source is clear in this animation from wikipedia:

Notice the waves coming in, well organized, not random.  I always want to discuss the source, not the slits.
Very cool stuff, Doug!!
Amir D. Aczel
Doug,
Reread what you wrote here:

http://www.science20.com/standup_physicist/blog/slit_experiments_and_coh...
"The logic of quantum mechanics in experiments with two slits is not just strange: the explanation using quantum interference is logically inconsistent. Constructive interference has "this plus that" making the big signal. Destructive interference has "this minus that" for the places with no signal. Yet one can use a super low intensity source, so low that "this never sees that". One cannot do the addition or subtraction. I accept the experimental results which have been confirmed time and time again. An explanation is a story we tell each other. The quantum interference story doesn't make sense."

The problem wasn't that you were trying to point out some expectations of the source in the usual two-slit setup, the problem was that you were claiming not only that interference doesn't make sense with single photons, but that it is not even possible as "one cannot do the addition or subtraction" because "this never sees that". You then go onto try to make it just about the source, to the point where when pushed you started including basically everything into your definition of the source.

Your article didn't even make sense. How did you remove interference from the explanation by using a coherent source? You act like you explained it an alternate way without interference. You didn't. There is interference in this experiment, even when the intensity is tuned really low. To claim otherwise is betraying a large misunderstanding of quantum mechanics on your part.

It is sad to see you give up on your blogging, but I guess you weren't gaining much from it anyway.

One reason for giving up blogging was replying to you specifically.  Take this statement:
There is interference in this experiment, even when the intensity is tuned really low
I know this.  I said this.  I have known that  for decades.  Yet you lay the claim that, well golly, I have a "large misunderstanding of quantum mechanics".  I have no intention of defending myself against either this false crime or my blog since because this is Amir's blog.

So to stay on topic, a super low intensity source with produces just one photon a year and which is described by a cosine function with a coherence length of one month will not show any interference pattern no matter how long one collect data.  A once a year photon described by a cosine function with a coherence length of a hundred years might start to show a coherence pattern after three years, but surely one would be obvious after fifty years.  The interference pattern will be lost if one collects data for a few hundred years.  The details of how the pattern changes over time could be modelled.
CR: "There is interference in this experiment, even when the intensity is tuned really low"
Doug: "I know this. I said this. I have known that for decades."

I know what you said because I can read it.
Anyone can read it. You did not say that.
Reread that paragraph I quoted.
You instead said interference does not make sense when the intensity is tuned really low. And not only does it not make sense to you, but you declare for these low intensities "One cannot do the addition or subtraction."

The problem is that you hold incorrect and contradictory information in your head, and when people point it out you get frustrated and claim we are making false statements. It's like your Z2 "numbering system" -- you kept switching between two different concepts and conflating statements ... it was contradictory and wrong, and even after Edward very explicitly wrote out all the math showing the properties and what exactly it was that you were conflating, you just refused to listen and continued to make claims that conflated the two separate mathematical structures. Even after David reiterated these points, it is not clear if you learned from it.

Doug: "I have no intention of defending myself"

Your instinct shouldn't be to "defend" yourself in this context. You shouldn't immediately conclude you are correct and need to defend your statements. Especially when many smart people like Edward, David, etc. are making the same points, you should be first considering if you made a mistake and try to learn from it. They aren't pointing out mistakes to "attack" you, they are trying to help you learn, as evidenced by the --numerous-- long posts with many details to teach you about a subject.

"Reread that paragraph I quoted."Here is a critical line...
"I accept the experimental results which have been confirmed time and time again."
That means that for a low intensity experiment, there will be an interference pattern seen.  That should be clear in my comment above about the once a year coherent photon source.  False accusations are exceptionally irritating.

"An explanation is a story we tell each other. The quantum interference story doesn't make sense."
It is this issue Amir is trying to address in his own way.  Comment on that, this is his blog, and stop trying to land a few last blows on my ego by appeals to whatever topic you feel like dragging in.

Here is an assignment for you.  Go to those old posts and only read David's posts for his style.  He doesn't throw in spurious insults.  "Large misunderstanding", "Go reread..." are insulting too, or perhaps you don't recognize it.  I had to do a lot of editing not to toss in my own insults.  Just so you know, I don't care about your opinion of physics or me.  At all.
Doug: Here is a critical line... "I accept the experimental results which have been confirmed time and time again."

You are not listenning.
Yes, your article wasn't claiming the experimental results were wrong. I know that, as did the other people commenting. The issue is that you were claiming the experimental results cannot be due to interference, as that doesn't make sense to you as you think there is nothing to interfere with when the intensity is low, then you proposed your own "reasoning" for the experimental results that fails to resolve any of the things you were complaining about.

This wasn't just a small typo, or a single badly written sentence, it was the entire motivation to set up discussion of "coherence patterns" instead of "interference patterns". Because to you it makes sense that "A coherence pattern can be constructed bit by bit", as opposed to interference because "One cannot do the addition or subtraction." with intensity "so low that "this never sees that"."

Do not let that distract from the issue with your statements on interference. Those issues are stark.

Doug: Go to those old posts and only read David's posts for his style.

To be fair, it was the shear persistence of Henry that finally got you to realize some of your mistakes in your gravity theory (theories). David instead tried to reiterpret things in a way that your statements could possibly make more sense, and in doing so saying things that you never originally meant and leading you on a path into more complicated math without ever resolving your issues. I would argue David introduced you to more concepts, but Henry actually helped you learn more physics than David. Each approach has its place, but trying to point out when you repeat old mistakes seems to be the best chance of you eventually acknowledging those mistakes and learning from them.

Doug: "stop trying to land a few last blows on my ego by appeals to whatever topic you feel like dragging in."

You were the one that brought this up, by _explicitly_ referring to the old discussion and making it clear you did not really understand the objections, nor did you learn. This isn't about your ego; no one is attacking you. People put a lot of time and effort into trying to help you learn various topics; this is just a reminder that you still need to learn from this particular lesson.

Doug: "Just so you know, I don't care about your opinion of physics or me. At all."

That is fine. I just hope you still care about learning physics.
If you respond again, and still don't understand what the commenters were objecting about, then I'll just let this drop for now and you can have the last word in case you still interpret this as something you need to "defend" yourself against.

I think it is wrong to debate a different blog within someone else's blog which is interesting in its own right (Topos theory, category theory, fuzzy logic).  That was in my first reply to CuriousReader which he ignored.  Feel free to delete this subthread since it just distracts from the topic at hand.  I don't mind if my comments are dropped, and I don't care about people who don't bother to log in.  That prevents both their editing their own comments and blocks back channel communications, both bad things.

One might get into a fun discussion about the difference between interference in a classical sense versus a quantum sense.  In what way does working with complex valued functions really alter classical cause and effect?  It might be fun to discuss, but not here where my words get twisted just enough to sound dumb.
You were the one that brought this up, by _explicitly_ referring to the old discussion
No, I made an implicit reference to the content of one of my blogs.  One was not obligated to read that blog in this context.   You provided the link.  You referred to the old discussion.  Another false accusation, so irritating.  Then you go through your accusation collection, including appeals to other critics that had little to no relevance to this blog.

I remain at peace with the idea that the oddity of the slit experiments depends on two things: the coherence of the source and that the math of that coherent source is complex valued.  Someday I hope to write some software that can demonstrate this is the case.

In the future, let people politely discuss issues, or just remain on the sidelines.
Doug:

You say (emphasis added):

I remain at peace with the idea that the oddity of the slit experiments depends on two things: the coherence of the source and that the math of that coherent source is complex valued.  Someday I hope to write some software that can demonstrate this is the case.

I advise you (and others) to not get too hung up on the use of complex numbers (this also relates to another message, below):  Any mathematical system that uses Complex numbers (or Quaternions, or anything else that can be formulated with the Real numbers) can be rewritten using only the Real numbers.

The principle utility of using Complex numbers is it can make the expression of the system more compact.  A secondary utility is in the body of mathematical work, using Complex numbers, that can be drawn upon.

David

Agreed, however “the body of mathematical work, using Complex numbers, that can be drawn upon” is quite over the top, the cup overfloweth. And really, the whole efficiency of doing a vector sum of quantum amplitudes first to get a resultant …. and only later taking the square to get real-valued probabilities … that all runs deeper than say …. the free body diagrams done in a first-year engineering course in Statics. And the fact that we can do so much with just a simple 2 in the exponent in essential equations …. to get back to real measurements …. that also stands out … yet of course, it couldn't have anything to do with Fermat's Theorem.

David, when are you going to complete your story on Tangent Spaces and metrics?

When different theoretical approaches give the same answers to a measurement, for example, then one learns to not attach too much realness or nearer-to-godness to the abstract objects that are being manipulated. A classic example is the Schrödinger versus Heisenberg approaches to quantum mechanics. Twenty years ago it was about how 5 different versions of string theory are “equal”. In Judeo-Christian terms, the Buddhist saying at the top of this essay is a caution against idolatry. Be that as it may, one is still struck by the unreasonable effectiveness of certain methods.

"blue-green" (aka Scott):

You ask "David, when are you going to complete your story on Tangent Spaces and metrics?"

I know...  I know...

The problem is getting a good, uninterrupted length of time to do the writing.  (Work has most definitely "doubled down" on me these last several months.)

Maybe this long(er) weekend will be helpful.

David

"blue-green" (aka Scott):

By the way, "the whole efficiency of doing a vector sum of quantum amplitudes first to get a resultant …. and only later taking the square to get real-valued probabilities …", "And the fact that we can do so much with just a simple 2 in the exponent in essential equations …. to get back to real measurements …", sort of breaks once we leave the scalar, non-relativistic "world" of "the Schrödinger ... approach to quantum mechanics".

Even just the Dirac equation makes such no longer nearly so simple.  Having non-abelian fields, such as SU(2) and SU(3), compound the issue.  (Dirac's bra-kets try to get one back to something that looks somewhat similar, but hides a great many, potentially important, details, such as the nature of the integral one takes in the process.  [I had to actually "back out" of much of the usual notation for my dissertation.])

David

I bring up tangent spaces and metrics because therein, one has bi-linear operations dealing with the contraction of two vectors into one of your beloved real numbers. The ubiquitous 2 (or 1-over-2) in the exponents is because one is dealing with two objects. ((I still sense Fermat's Last Theorem and Pythagorean triplets lurking in the undercurrents.))

And then you bring up Dirac and the strange objects with which he was wrestling (even if one can rather meaninglessly stretch them out on a real number line). Sure, complications arise, for the very reason that the truer objects are not real numbers. Be that as it may, unitarity is preserved. From your response, one might think that the fundamental unitary and linear rules of quantum mechanics have been obsolete since Dirac. Is this the impression you want to leave with readers? Or is this about Lebesgue?

"blue-green" (aka Scott):

Unfortunately, I'm not quite certain what you are getting at, or what your objections are to what I have stated.

I brought "up Dirac and the strange objects with which he was wrestling" not because of any "fundamental unitary and linear rules of quantum mechanics [that] have been obsolete since Dirac."  I brought such up because they illustrate my point:  There are other, additional ways of obtaining "objects" that cannot be "stretch[ed] ... out on a real number line" without ever having to resort to Complex or other number systems.

(Incidentally, all non-abelian [Yang-Mills] fields involve non-linear systems of equations.  Indeed, the same is true of the Higgs field, since, otherwise, there would be no way for it to provide the necessary symmetry breaking.  However, that's beside the point I was trying to make.)

The "fact" "that the truer objects are not real numbers" does not, in and of itself, imply any necessity to use Complex, or any other number system, unless and until one has a more fundamental "grasp" of what "the truer objects" actually are.

What I'm advocating is not a "love" of "beloved real numbers."  What I am advocating is to not get too caught up with any particular representation.  It's quite akin to your comment:

When different theoretical approaches give the same answers to a measurement, for example, then one learns to not attach too much realness or nearer-to-godness to the abstract objects that are being manipulated.  ...

So, I think we agree at a very fundamental level.

David

P.S.  The comment I made that I think sparked your question

...  From your response, one might think that the fundamental unitary and linear rules of quantum mechanics have been obsolete since Dirac. Is this the impression you want to leave with readers? Or is this about Lebesgue?

was

...  (Dirac's bra-kets try to get one back to something that looks somewhat similar, but hides a great many, potentially important, details, such as the nature of the integral one takes in the process.  [I had to actually "back out" of much of the usual notation for my dissertation.])

Well, what does <ψ|ψ> actually signify?  It looks rather like |φ|2 (= φ*φ), doesn't it?  Especially if one thinks of <ψ| as something akin to |ψ>*.

On the other hand, |φ|2 is still a function of space and time.  (Isn't it?)  While <ψ|ψ> is simply a single real number (or, at most, a real function of time), since it involves an implicit (spacial) integral.

Of course, this is to say nothing of the justifications in terms of "Hilbert space", and such.  (We're simply taking the inner-product in the Hilbert space.  Aren't we?)

I won't go into any further details here, except to say that the nature of the spacial integral is not as "flat" or "rigid" as one may believe, based upon (Special) Relativistic Quantum Mechanics (and the spacial nature of Inertial Reference Frames within Special Relativity).  It is far more flexible, and, thus, much less strictly determinable.  (However, for any operators satisfying the right conditions, relating to the Dirac equation, there is a very nice [and, dare I say, essential] invariance.)

I don't want to keep you from other duties and from rebooting your article on tangent spaces and metrics. Still, I'd like to take you up on your dare. A simple yes or no can suffice as to whether you think the unitary basis of current quantum mechanics is here to stay. I have thought that it relates to an invariance which could be called “conservation of information”, however, even though QM has been interpreted by Zeilinger and others as being a theory or information, I feel that you will not be happy with your “dare I say, essential invariance” being related to “conservation of information.” Too many definitions to sort out ....

Frankly, I am just trying to encourage you up to complete your original task on metrics from which the essential invariances will fall right out, pair-wise or otherwise.

Anything that walks, talks, and swims like a duck can be called a wombat.  Rewrite it so it uses only real numbers and behaves exactly like complex numbers should you choose.

I am unable to find the reference (my papers are not so well organized), but I am trying to quote Stephen Alder who was in turn quoting from Birkhoff and von Neumann's work on the logic underlying quantum mechanics.  The gist was that one could do quantum mechanics over the real number field, but one would not see quantum interference.  To be able to calculate quantum interference requires a system that has the mathematical properties of complex numbers.  Those guys were able to make the point much better than I.

This is my own spin on the issue, so no need to pour salt on the snail.  It is not that complex numbers are compact.  It is that complex numbers are a mathematical field that is not totally ordered.  Real numbers are a totally ordered set, one can always say this number either came before that one, or after that one, or is that one.  No other interpretations are possible for a pair of real numbers.  While one can invent a way to order complex numbers, that then depends on the ordering system used.  A different ordering system will line the ducks up a different way.  Quantum systems are not random in the traditional "roll those dice" kind of way.  The functions used to describe quantum systems are quite exact and exacting.  The issue to me looks like they form a not ordered set of data which is a fundamental property of complex numbers.
That sounds good and relevant.
Doug,
I like your papers on using quaternions--step up from using C. Have you seen the article by John Baez in the Bulletin of the AMS (April 2002, pp. 145-206) on octonions? He gets to a deep comparison of using R, C, H (quaternions), and O (octonions) in mathematical physics. As you go up and lose structure (as you wrote above, C is not ordered, and then you lose commutativity, and finally associativity) the use of these structures becomes mostly algebraic. If you haven't seen that article, look it up--he even explains some string theory in this context.
Take care,
Amir
Amir D. Aczel
Wow, the pdf is free!  The reason I have not dived deep into octonions is in the introduction:
However, there is still no proof that the octonions are useful for understanding the real world. We can only hope that eventually this question will be settled one way or another.
I have listened to Baez sing the praises of Bott periodicity before.  It is no doubt real and important in some way, but I quickly get lost in the alpha-numerics of group theory.  I feel quite differently about the symmetries U(1) and SU(2) now that I have animations for both (and even that was not easy, needing a correction or three from David Halliday).

Doug:

You say that the "gist" you gleaned from "Stephen Alder who was in turn quoting from Birkhoff and von Neumann's work on the logic underlying quantum mechanics" was "that one could do quantum mechanics over the real number field, but one would not see quantum interference."

I'm reasonably certain that they did say something that may be interpreted that way, within the context of their statements.  The problem is that within the broader context of my statement, this is no longer true.  One can fully formulate "the logic underlying quantum mechanics", complete with "quantum interference", without ever even mentioning, let alone using complex numbers.

However, along the lines of your "own spin on the issue", the structure will still contain the underlying structure of the complex numbers embedded within it (though such could, conceivably, be well hidden).  For instance, rather than a "plain Jane" inner-product, one will have a somewhat more complicated system of two Real equations, or a matrix equation.

Additionally, there are other ways of obtaining unordered (or semi-ordered) sets for use within Quantum Mechanics (QM) than turning to Complex numbers (or Quaternions, Octonions, etc.).  As soon as QM made allowance for the introduction of groups/algebras (and even before that with operators and the Pauli or Dirac "matrices") one has all such, and more at one's disposal.

Besides, even with Classical Mechanics (all based upon Real numbers [though occasionally taking advantage of formulations using Complex numbers]) we already had unordered (or semi-ordered) sets.

Of course, this is not to take anything away from “the body of mathematical work, using Complex numbers, that can be drawn upon”, which, as blue-green has stated, above, "is quite over the top, the cup overfloweth."

David

I need to read this more carefully, but what immediately struck me is that I've been considering a 4-value logic system for Prolog programming as an alternative to Prolog's 'negation as failure', in which something is ether considered provable or non-provable as derived from a knowledge base of facts and rules. These possibilities would be 1) A is provable. 2) the explicitly stated negation of A (exNot A) is provable. 3) Neither A nor exNot A are provable. 4) Both A and exNot A are provable. This would also seem to be applicable to RDF & the semantic web, which must be robust to contradiction.

Is the relation of the observer to the eigenstate (as in Relational QM) a topos? If so, why Is topos exclusive of many worlds?

Hi, The topoi are alternative mathematical universes within which you could view physical systems and phenomena. I don't know how you view "many worlds" within such a system (where are these "worlds"? and what is the cardinality of the set of worlds?--is it that of the rationals, or of the continuum? or of what?). The problem of QM and the mind was investigated and championed by the late John Archibald Wheeler of Princeton (I believe that Everett, of "Many Worlds," was his student). Roger Penrose writes about physics, math, and the mind as well. Generally, I believe, physicists try to distance themselves from such assertions as one being able to affect the behavior of a quantum system by interaction with the observer's mind; although, of course, it is hard to find an explanation for why a particle will choose one path or another when an observer can distinguish paths, and take both paths when s/he cannot... It's a very big problem in the interpretation of QM. Where a topos might lead vis-a-vis this thorny discussion is anyone's guess. Hope this helps!
Amir D. Aczel
My question simply related to the double vs. Single slit experiment. The choice of slits requires external interaction (the observer), and each outcome becomes a path (or reality in many worlds). Using Topological (Topoi?) relations for the system with both outcomes just seems a natural step (like a Feynman diagram), although I am not familiar with this annotation and maybe this would be a misapplication.

Not sure how groundbreaking it is to suddenly become aware that not all systems are binary in nature. For example, with a dimmer installed my light bulb is just not off or on but can vary in brightness from 0 to a maximum value - it's called a continuum. In fact, on/off, yes/no, 0/1 binary systems are actually quite rare though, of course, as a precept , the logic of such a system underlies digital computing. I'm neither a physicist or mathematician but would be shocked if the theory of topei is the first to address systems containing a wide diversity of possible values. So what's germane here?

Well, it's certainly not the only system. The whole of probability theory rests on such assumptions (and the mathematical theory of measure is it's theoretical underpinning). There is also fuzzy logic; and see Doug's comments above. The novel thing, really, is that topoi are a good conceptual framework in the sense that they naturally create mathematical universes that intrinsically have non-zero-one logic to them.
Amir D. Aczel
Hello Everyone:
I am not an expert on topos theory, but Professor Linton, whom I quote in the article, is a prominent logician. Here are his comments to me in an e-mail I received from him right now. I think that they answer some of your questions above more completely than I have:

"Let me just offer another timely "neither true nor not true" illustration:
the assertion (in computability theory) that P=NP (or its negation, matters
not to me here): at the moment, in our present state of knowledge, that
surely qualifies as neither true nor not true, n'est-ce pas?

Or again, from Section CXCVI of Santi Parva, part 2, of the Mahabharata
(in one of the Roy/Ganguli translations): > ... are both concerned and again unconcerned (with silent recitations).

Another pair of thoughts that blog piece triggered, as regards the
replacement of binary Boolean logic with probability: 1st, that the
spectrum of probabilities interpolated between true = 1 and false = 0
is analogous to interpolating a scale of grays between true = 1 = white
and false = 0 = black; and 2nd, that the contrast between a probabilistic
extension of logic and a more general topoidal one is parallel to that
between the one-parameter gray scale color spectrum and the full RGB
color spectrum (of which gray-scale is the special case with R = G = B)."

(Thank you, Fred!! Amir)

Amir D. Aczel

Very nice article. Sounds like material I have been playing around with for a long time, even the indirect reference to Jesus coming with a sword to separate the sheep from the goats with lots of spilt blood and no excluded middle …. a recipe for intolerance if there ever was one.

Category theory I remember being in a stand alone chapter at the end of my hefty graduate level text on Group Theory. Your square diagram showing 4 transformations is the essential diagram of what higher mathematics is all about. (However, I would have used arrow directions that make it easier for the eye to run around the diagram or get to the same place via different routes. The touchstone for measuring curvature and gravity is the degree an object is changed after it is parallel transported around a diagram like this.)

Leonard Susskind's way to express quantum logic is that it does not follow the usual exclusiveness and inclusiveness of Venn Diagrams. In my own words ...

“quantum mechanics is silent about a great many things, including what color of shirt I am going to wear tomorrow. It is not anywhere as tight and restrictive as is classical physics. We see that in its fundamental admission of random processes .... and in its use of a type of logic which isn't the usual Venn Diagram logic you learned for classical realism. QM has a different methodology for common words like AND ...and OR. Its Hilbert Space math with complex-valued vectors is not just an extension of ordinary vector analysis. The objects in it are different and behave differently.”

The basic unit for the fuzzyness is the qubit, a complex-valued state vector on a unit-sphere ... which is not the same as variations of grey on a dimmer or the two-dimensional settings of a joy-stick. You have to get to the root of why we have complex numbers in QM. Wheeler's student Wooters claimed (in his thesis) that it has to do with measuring distances and “nearness” in Hilbert Space.

Hi Blue-Green,Thanks for all this!! Sorry about the diagram: I should have drawn my own (a triangular one to illustrate the actual point I was making) but instead looked for a "free" one online--from Wikimedia Commons...and that was the only one there. Thanks for the comment on color vs. grey--I'll tell Linton.
All best,
Amir
Amir D. Aczel

The essential diagram of higher mathematics has 4 “regions” and 4 “arrows”, as in the following example.

Let the bottom two regions be called Q and A to represent a place from which you are going to ask questions and a place to from which to find answers. An arrow F from Q to A represents either a particular map from a single question to a single answer or entire bundles of questions to answers. In engineering and real life, answers to even simple questions (like those from number theory), seldom come in a single quick step or arrow. One has to first recast a problem in a new space which is amenable to solving the puzzle. You step up to a higher abstract space which I'll call HQ. The symbol for a map (or maps in general) from Q to HQ is an arrow T, which Transforms  or Transcends (at least some of the) elements in Q to elements in the “higher space” HQ. If a problem can be solved in the transcendental space HQ, then there is a solution indicated by an arrow HF from HQ to a point in an answer space HA. To bring the answer back down to the level of the initial question, one needs an arrow from HQ to Q, which I'll call P for Prometheus or Projection.

Calling one level of the diagram higher than the other is a matter of convention. Some people, ancient ones in particular, picture the process as going deeper or underground, in which case, one may want to flip the whole diagram and replace the H's with D's.

Thanks, Blue-Green. A great reference on this is Robert Geroch, "Mathematical Physics," U. Chicago Press. The whole book uses category theory to do math physics (wonder he can do it!!)
Amir D. Aczel
I enjoy your company.

Me too, David!
Amir D. Aczel