Quantum mechanics is natural, not plagued by problems; maths relevant for physics worships the rules that are natural for continuous, not discrete, structures
I received my copy of Scott Aaronson's book – thanks to him – and I sometimes find some time to read a chapter or two. So let me post several comments on my impressions – and generalized comments about thinkers like him.
First, the book is very witty, narcisistically witty. Its language is very informal. It's totally OK with me because at least onehalf of my parental background is extremely detached from any world resembling the Academia (and even rural if one returns by one more generation). However, the averagemanonthestreet language sometimes sounds really bizarre and astroturfal if you realize that the author is a archetypical model of a leftwing Academic carefully parroting all these people's collective delusions – including "gems" such as the very global warming hysteria – and confined to a snobby ivory tower. ;)
The preface tells you a funny story how Scott Aaronson didn't sue the authors of a commercial who quoted a few sentences from his talk or whatever it was. But it already presents some "basic scientific philosophy". The only problem is that most of the basic theses are wrong and even when it comes to the correct ones, Scott talks the talk but doesn't walk the walk.
Now, after I have finished reading Chapters 13 as well, the preface looks much less agreeable to me than it did before, probably because I see what was actually the "grander goal" of various sentences and I can evaluate in a more "moral" way than just by their literal validity.
Right after I finished the preface, I thought that the comments about the author's "suspicion about everything that is continuous" were the only major problem. But right now, it seems that most of the basic philosophic tenets are deeply flawed.
For example, the counterintuitive nature of quantum mechanics is already discussed in the preface. And if I realize what he actually wrote, it seems so fundamentally flawed to me! First, Scott wants to make as "modest" claims about the interpretations as possible. So he announces he won't pick any preferred one but he will agree with anyone who says that there is a problem with quantum mechanics and disagree with every interpretation that presents quantum mechanics as a complete and consistent framework to do physics.
In most respects, I would also prefer to be independent of "tastes" and illdefined words and philosophies that don't produce in principle observable differences from others but the main message that goes beyond these things would be exactly the opposite one's than Aaronson's. There isn't any problem with quantum mechanics. If you want to find a problem with a theory within the scientific method, you must actually show what it is – its incorrect prediction of an observation or an internal inconsistency. These people's vague feeling that "something must be wrong with a theory" even though they know very well that there doesn't exist any sensible observation that would actually back this criticism only shows that these people's thinking is unscientific. They prefer emotions, feelings, animal instincts, dogmas, and stupidity over cold reasoning and empirical evidence.
The problem gets even more manifest when the author says what we should do when we find out that our intuition was wrong. We should fix our intuition to be sure that next time, we will have more sensible expectations and what Nature is telling us won't look crazy or extremely unlikely to us again. I originally thought that this was what he was writing and I would agree with that.
However, now I think that this "right reaction" was just the beginning of his explanation that ended up with a totally different conclusion. He thinks that he was told that his physical intuition was wrong while no one gave him a path to correct his intuition. It's like flunking him on an exam without providing any hint how he could have done better, Aaronson wrote. He thinks that it's sensible for him to switch to other courses where he can earn As.
But whether someone told him how to correct his intuition, the most important thing is still that he has failed the exam! His intuition didn't agree with Nature's actual behavior so he and his musings had to be flunked. The purpose of science isn't to provide help to children who were left behind. The purpose of science is to find the truth by eliminating the ideas that are wrong. In Aaronson's analogy, flunking the bad students isn't a negative side effect of science; it is its very purpose! And I think that much like the Islamic terrorists, he is personifying the wrong ideas about quantum mechanics – they become a student in his metaphor – in order to convert such a student to a human shield whose destruction should be considered inhuman. But it's still totally necessary to destroy what is behind the student if we want to study these things scientifically!
Now, it's sensible for a flunked student to try something else (falsified ideas in science usually get no additional opportunities: that's a difference showing why the personification is misleading). But it's still important that if he hasn't found a way to correct his knowledge or intuition, he has not mastered the subject. In these circumstances, it's kind of painful for him to write a book about the subject he has not mastered! Claiming that the exam was hard, the examiners were cruel, and they didn't provide the poor whining student with any hope how to get better shouldn't be used as an excuse to write rubbish about things he doesn't understand.
Incidentally, this particular opposition to science – "science must be corrected so that no poor child is left behind" – is a predominantly leftwing assaults against the dignity of science (I count spoiled whining frats who flunked an exam among the leftists). The association of the phrase with George W. Bush is an exception that confirms the rule.
May we help the students to understand how to fix their intuition (about quantum mechanics and, more generally, about other things)? I have tried to do so for years and the results are mixed. But these are mere pedagogical efforts, not a part of the scientific research. Even if almost no one can be explained how to correct his intuition, it means absolutely nothing for the validity of quantum mechanics. The validity of a scientific theory doesn't boil down to the science's being accessible to most people. Indeed, most of the modern science – and not just quantum mechanics – is pretty much inaccessible to most people in practice. But that doesn't mean that there's anything wrong about the science! To show that there is something wrong with a scientific theory, one must find valid evidence of an internal contradiction or a contradiction with the experiments according to the proper professional rules of science, not just to whine that these things are hard to understand!
At the end, the scientific claims on quantum mechanics in the book (so far) are bad but they aren't that bad. He wants the information to become the central player. It's just fine with me. It's an OK sketch of the philosophy predestined as a theme for a book. However, if such a meme is promoted to a key dogma that decides about the validity of all other detailed scientific questions, you shouldn't be shocked that one ends up with wrong answers to most questions. Science can't be guaranteed to obey predetermined dogmas. No one can guarantee that if you try to write a book where a whole scientific discipline is presented as a consequence of a short slogan, it will be a valid book. Indeed, most likely, it will be garbage.
Chapter 1 focuses on Democritus, atomists, and some very general philosophical questions about experience vs abstract scientific theories. But they aren't really brought to the level of the stateoftheart science. So in some sense, it remains a superficial chapter about the ancient Greek philosophy that is disconnected with the rest.
Chapter 2 talks about sets, their elements, axioms, quantifiers, axioms of Peano Arithmetics, cardinals, ordinals, Axiom of Choice, continuum hypothesis, models for axioms, and general observations about the consistency of axiomatic systems. It's nicely written but given the fact that the book is supposed to be primarily a book about a discipline of physics, I would say that this chapter is pretty much disconnected from the primary topic of the book. The questions related to set theory aren't "remolded" to become useful for or valid from the viewpoint of physics. The most likely reason is that the author just hasn't understood what the relationship is. So he's only eclectically combining things he has heard from different sources.
What's the relationship?
Axioms and set theory may be used as a starting point to define the mathematical frameworks used in physics. But axiomatic systems and physics aren't the same thing. The most important difference between mathematics and natural sciences is that in mathematics, one may study pretty much any sets of axioms he finds interesting; in natural sciences, we want to learn the true axioms that are respected by Nature, at least within an approximation.
I am afraid that Scott Aaronson – and pretty much everyone who is trying to reshape physics as a discipline of computer science – just fails to understand this basic fact about the scientific method. He apparently thinks that if he writes down some facts about the integers or discrete sets or Peano algorithms or algorithms, they – because they look "elementary" or "fundamental" from some viewpoint (as starting points, they contain no reducible beef in them) – must be fundamental things behind physics, too.
But that's now how natural sciences work. Peano Axioms or algorithms or Turing machines may be the elementary concepts in various branches of maths and computer science but they're not guaranteed to be – and they actually aren't – the key ideas upon which the structure of physics is built. In science, we must be openminded what are the right axioms – even what is the "general spirit" of the axioms, the philosophy of the right theories, and the character of the mathematical structures that are used to describe the world. Those things simply aren't up to us.
Scott Aaronson and pretty much all the computer scientists trying to conquer physics, "discrete physicists", and related sets of people just can't accept this basic point.
They pompously think that they're in charge, they may order Nature to follow their dogmas, and it's Nature's duty to find a way to follow them (and to pay income taxes, carbon taxes, and 50 other types of taxes as well). But Nature won't accept this role. Nature is a harsh examiner who evaluates Scott Aaronson's predictions and – more generally – intuition. She finds out that he fails in Test 1 and Test 2, among others (he thinks that Nature should be discrete etc.), so she flunks him. When he starts to whine that Nature is obliged to help him to get a chance to get an A, She just tells him to sh*t up, f*ck off, and, if needed, hang himself. He has no f*cking business to tell Nature how She should behave or even to promote the culture of entitlement or other leftwing diseases and corruption.
Chapter 3 reformulates Gödel's Completeness and Incompleteness Theorems in terms of (hypothetical) algorithms that either stop or not or decide whether another algorithm stops or not. By a trivial "I am a liar" argument, such an algorithm can't exist which proves the Incompleteness Theorem. I totally agree that the computer scientist's way of thinking about such things simplifies much of Gödel's thinking tremendously. Turing gets some credit here. And Aaronson talks about the consistency of axioms supplemented with additional assumptions about their (in)consistency. It's all nice. But again, these things haven't really passed the tests of "validity from the viewpoint of physics".
A physicsoriented question appears at the end of Chapter 3: Are we really talking about continuous objects themselves or about finite sequences of symbols that talk about continuum?
That's a good question and I am inclined to say the latter. If we talk about specific things, these specific things are always countable because they must be describable by a finite sequence of symbols. Even when we talk about intervals of real numbers that arguably contain an uncountably infinite number of real numbers, we must still specify the endpoints of the interval by a finite sequence of words or symbols – and such sequences are "discrete" i.e. "countable".
This is why I tend to consider the very fact that real numbers are uncountable to be nothing else than a linguistic curiosity: the actual, welldefined real numbers you may ever encounter form a countable set! This is why the uncountability of the real numbers – and the whole discipline of maths based on this formally provable claim and similar claims – doesn't have implications for "talking about physics".
On the other hand, the fact – let me now fully assume that I do take this position, and I mostly do – that we only talk about finite sequences of symbols does not mean at all that discrete maths and its axioms should be the foundation of physics. Even when I say that we are talking about finite sequences from a discrete set, it's still important what we're saying about them. And to do physics, we should organize these sequences and "discrete objects" in such a way that they talk about properties of continuous objects because those are ultimately fundamental in physics, as we know because of many general reasons.
So talking uses finite sequences from a countable set but in physics, it still matters what we are saying, and if we're not saying things about intrinsically continuous structures and properties that continuous structures may have, it will be either invalid or nonfundamental as chatter about physics!
I hope that the dear TRF reader crisply understands the difference between the statements "all language and communication uses discrete symbols" (true) and "Nature is fundamentally based on axioms describing discrete mathematical structures" (false). But I do feel that Scott Aaronson must be overlooking this notsosubtle difference. It reminds me of Amy Farrah Fowler's silly comments to Sheldon Cooper when she suggested that neuroscience was more fundamental than theoretical physics because she may understand the functioning of a brain that is discovering an important theory in physics. The problem is that while she may, she doesn't. To find a right theory is a task that can't be reduced to the understanding of the functioning of the brain; after all, she hasn't completed the task. Similarly, theoretical physics can't be reduced to grammar, linguistics, and the history of languages because it still matters what the physicist is saying and a linguist has no idea about that – that's why linguists don't have any credentials to claim to master theoretical physics, either. I could add a few similar examples with additional occupations. Secretaries and bakers also provide something important for the physicists but they are not physicists themselves. My broader point is that it is comparably silly to start with Peano Arithmetics as the foundation of physics as it would be to start with some basic rules of grammar or linguistics. They may be the natural starting points in a discipline but the discipline just isn't physics. What is true and fundamental in physics is decided according to the rules of physics, not according to the rules of bakeries, bureaucracy, linguistics, or set theory.
At the end, pretty much all the silly "conceptual" things that people (not only Scott Aaronson) are saying about the foundations of science boil down to the same problem: these people just refuse the scientific method as a method to verify the basic tenets of their belief systems. From this viewpoint, some people's belief that alternative medicine or homeopathy must be much more potent than just the narrowminded modern Western medicine is based on the same "intellectual mistake" as Scott Aaronson's belief that continuous objects or quantum mechanical postulates are "suspect". It's a belief. They aren't willing to put this belief at risk and test it. They aren't willing to find out whether it is right. They aren't willing to learn that their belief systems are actually based on tons of stinky feces. They prefer to believe rubbish and search for allies and excuses explaining why flunked spoiled frats should still write lots of things about things they haven't mastered.
I may continue with Chapter 4??? sometime in the future.
Monday, April 22, 2013 ... /////
Scott Aaronson: a prototype of some confusion of IT scientists about physics
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Luboš Motl
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snail feedback (34) :
I like your honesty. I was going to ask how you get away from the fact that our description of physics must use a discrete set of symbols, but then you already addressed this question. You state "continuous objects because those are ultimately fundamental in physics, as we know because of many general reasons.", I think I must learn some string theory before I can appreciate those general reasons. But in any case, physics is a mapping from a discrete set of symbols to a discrete set of observations, what comes between, what the symbols talks about, must be a matter of interpretation? Maybe I am oversimplifying
Dear W, thanks  I have no reason to get away from something here. The fact that we're using discrete symbols to communicate ideas seems to be as important for understanding how science and language etc. works as the importance and properties of continuous mathematical structures is important for understanding physics. These two things just don't contradict one another.
I don't think that you need to get up to string theory to understand the importance of the continuous starting point. Quantum field theory is surely enough. Its equations of motion  or let's talk about the action  continuously depends on continuous fields and continuous parameters. And predicted quantities  essentially probabilities  continuously depend on the properties of the environment (essentially a part of the question: momenta of the particles), if these properties are continuous, as well as on the continuous parameters in the Lagrangian, and discretely depend on discrete data but all the discrete properties of fundamental objects  such as the projection of the spin  are emergent side effects of some continuous underlying physics if one looks deeply enough. In this sense, you need to understand string theory to see that the spin as a property of a particle isn't an ad hoc discrete addition to a particle's properties but a consequence of something deeper that is constrained by some continuous rules  continuous equations relating a priori continuous objects where all the dependence is essentially continuous.
It's OK to describe the laws of physics that "contain the Nature's wisdom" as a finite sequence of discrete symbols; and you may even say that the observations are discrete sets of symbols, too. And yes, physics is the mapping in between. But physics must be the right mapping and the right mapping just uses the continuous mathematical structures pretty much everywhere. If it doesn't, it's not right. That's not an a priori obvious dogma or some political opinion or mine; it's something we've learned by studying Nature scientifically. Everything we learned are "some details about the things in between". If someone is unconstrained by these things and only looks at the "end points", he may conclude that everything he sees about physics is discrete. But that because he overlooks *all of physics* completely! It's because physics *is* about all the properties of the mapping, the right way to connect the discrete endpoints, how to connect the ropes and what gadgets and "black boxes" have to be added to do it right. And these connections of ropes and gadgets just have properties that contradict virtually everything that the "discrete physicists" want to impose on Nature.
I didn't quite understand what you say about what "must be a matter of interpretation". The interpretation of symbols we use in physics is important but what's important is that some interpretations (usually at most one) is right while all others are wrong. So if something is about the interpretation, it doesn't mean that it leaves room for subjective choices.
Aaronson's failures to understand quantum physics in general do not seem to affect his understanding of quantum computation, because he has been a top researcher in that field. However, they give him curious blind spots. He is morally sure that the power of quantum computers as he defines them must be exponentially better than classical computers for some problems in between P and NPcomplete, but not helpful in solving NPcomplete problems. (This is very apparent with his recent work on linearoptical systems, because it leads him to underplay the significance of his own results!) He also never considers the possibility that the simple components from which all the "quantum computers" he analyzes are built might not exhaust the possibilities for computing functions by experimenting with quantum systems (the proofs of "universality" of the standard model of quantum computation are naive and circular).
Right. I just want to make it clear that I don't think that Aaronson is among those who misunderstand QM most.
It's really about the key that he is working on an applied physics problem where he's given a Hamiltonian  described as a discrete matrix on a finitedimensional Hilbert space  and remains (possibly so, without a professional impact) unaware about the question what is the right Hamiltonian that describes such a quantum computer in the real world, what the stuff is composed of, and why it has the properties it has.
So he ends up with some naive extrapolations of his computational engineering  which is effectively some discrete effective description of some degrees of freedom  and assigns them much more importance in the scheme of things. It may have some impact even on the "applied science", perhaps even negative one, but the sign is probably random.
Really enjoyed this post! I'm only a linguist and certainly cannot add to the progress of theoretical physics for sure. but, I always thought Amy's statement to Sheldon was totally lame about neuroscience trumping physics.
Thanks, Ann! The dialogue was added for a comical effect and stems to the idea that she is the ultimate female peer of Sheldon on the show. Of course, if there's no isomorphism, there's no isomorphism.
Amy's opinion is probably not too widespread  it's more of a children's pissing contest about favorite subjects at school.
Narcisistically witty, that's correctly said. SA belongs to the group of researchers who use their talent and intelligence not to really advance science & knowledge but rather to promote themselves and show off. I regularly meet him and his peers at various QIT conferences and it's a pain to watch them: how they need to underline their existence  an intolerable odour of egos with zero substance. People like SA, Lloyd (to name just a few) are mere poseurs as I already wrote before. QIT community is surpisingly full of this type of people. And I think your halfreview only confirms this impression.
LOL, can't wait to find out what you think of the rest of the book!
But one need go no further in QM than chapter I of Dirac for SA to be inconsistent with QM was about how I too had read some of your blog remarks. Thanks for clarifying.
I met a building sized computer in my teens and .... Well I am now forewarned but comfortable to buy his book in spite flaws,
Does it really matter that the numbers used in theoretical physics are real as opposed to rational since the rational numbers are close enough to being continuous for all practical purposes? Can we rule out nature being rational in that sense?
Dear Scott, it's good for you if you think that we agree but Nature is continuous not only at the level of probability amplitudes  they have to be continuous  but also at the level of elementary observables. They have to evolve as functions of continuous time.
They may have a discrete or mixed spectrum but observables such as x,p have continuous spectrum and it's inevitable for a theory with Heisenberg equations that depend on the continuous time that observables with a continuous spectrum exist. In fact, their existence is necessary for the existence of constraints that determine or partially determine the theory  such as continuous symmetries.
It's also untrue that the holographic principle suggests that quantum gravity has discrete degrees of freedom. It only says that the entropy associated with a finite region is bounded. But it is not expressed in terms of any qubits  the localization to discrete units doesn't hold. It's always more correct to think of information in physics as being composed of nats  with the base equal to e=2.718...  and not bits! All physically meaningless formulae for the dimensions of large Hilbert spaces are given by exponentials of natural exponents and not by powers of two with simple exponents!
Two quick responses:
1. If you have a source with finite entropy, then a sample from the source can always be approximated to arbitrary precision by giving a finite sequence of bits. I suppose it could also be approximated using a finite sequence of "nits," if there were such a thing as a nit.
2. This isn't a negative comment about neuroscience, but to compare the level of generality of neuroscience to the level of generality of theoretical computer science indicates an extreme lack of familiarity with one or both fields. Suppose Amy FarrahFowler wanted a theory that described the commonalities, not only between human and monkey brains, but among ALL animal brains ... and better yet, among all brains of all the *possible* creatures that could arise by natural selection in our universe ... and better yet, among all computational artifacts that could possibly exist in our universe ... and even better yet, among all computational artifacts that could possibly exist in any conceivable universe that was remotely like ours. What would she be led to? Answer: the theory of computation.
Would not she be led to conclude that a Turing machine fails to provide for superposition and uncertainty? how can any deterministic model can ever sufficiently model a system that is inherently not consonant with a deterministic foundation?
Well I shall read the book. I am sure that I am too ignorant on this debate.
A Turing machine can certainly simulate quantum mechanics, albeit with exponential slowdown. Whether it can do so *without* exponential slowdown is an open question  one of the biggest at the intersection of physics and computation. We conjecture that it can't, but proving it will require major advances in theoretical computer science. Want to know more? Read the book! :)
"But that's now how natural sciences work." No. Not now.
What happened to the dashing Scott Aaronson who conducted the sublime soapbubble experiment? That one was younger. The current one is older. Some people don't age well. They begin recycling. It's sad.
Lubos, this is a very deep issue and I am glad to hear that you are saying the same things I say to other mathematicians about this. The fact that inputs, outputs, and descriptions are discrete does not mean that the continuous math can be eliminated. Although there are alternative ways to get the continuous math than the standard textbook route through ZFC, you can't avoid infinite sets. At the same time, uncountable pathologies like the BanachTarski paradox seem to be irrelevant to physics.
I'd like to understand how much math, axiomatically, is actually needed to fully support quantum physics. For example, everything can probably be done in some finitely iterated powerset of the integers; you need reals which are like sets of integers, functions which are like sets of reals, operators which are like sets of functions, and a few more levels, but you don't even need infinitely many infinite cardinal numbers, and you probably don't need the Axiom of Choice either.
In my Ph.D. thesis I showed that Pitowsky's "hidden variable" theories, which got around the Bell inequality by violating Fubini's theorem with nonmeasurable functions where changing the order of integration corresponded to measuring observables in a different order, needed axioms going beyond ZFC (it was consistent that no such functions existed and Pitowsky's use of the Continuum Hypothesis to construct them was therefore not eliminable). From that experience I developed a deep distrust for applications of "pathological" sets to physics. But articulating exactly what kind of math is "pathological" is difficult.
What are your thoughts on this?
Please read and report on the rest of the book. ;)
"...We don't have zillions of other inequivalent concepts of computability that make just as much sense; in fact, we don't have even one such concept. " I disagree, you are ignoring a possible weakening of the Tennenbaum theorem. For example what happens when isomorphism relation used is not exact. Recursivity can be relative.
You could extend that to other branches LM say make a computer model of world climate based on radiation forcing and then try making real world accurate predictions. It really isn't any different to what SA is doing like all computer models they are "somewhat useful" the problem comes when the users forget it is a model and they have no idea of understanding the reality of the system.
Dear Luke, the rational numbers' being close to any real numbers  yes, there are arbitrarily close rational numbers blah blah blah  is not a sufficient condition for the rational numbers to be sufficient in physics.
What's needed in physics is not so much the proximity which is a stupid triviality. What's needed about the real numbers is the continuity and things that can only be done with objects that depend on continuous numbers  such as differentiation. The evidence is overwhelming that these mathematical structures that can't work with rational or other discrete numbers  complex numbers; real numbers; differentiation; integration; functional analysis, analytic extrapolation; smooth manifolds; fiber bundles, and so on, and so on  are used pretty much at every place of the fundamental laws of physics and one can't understand any physics without these continuous mathematical structures.
Dear Robert, what I don't understand is that you talk about the research of Turing machines. But physics doesn't find answers by observing and studying Turing machines; it studies Nature around us. These are completely different things, aren't they?
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Lubos, your inability to suffer the possibility of any useful perspective besides your own remains legendary and comical. Most of us eventually outgrow the undergrad pissing contest about "whose field is best," when we realize we went into whichever field we did because it seemed deepest to us, but other people went into what seemed deepest to *them*, and it's best that we didn't all feel the same, because this way we all get to learn from each other.
Indeed, your spittlelaced attacks on the "reference frames" of mathematicians, computer scientists, and just about anyone else make the very title of this blog a cruel irony (intentionally or not, I can't decide!). The whole point of SR was that we don't need a single privileged frame, as long as we can all translate our observations into each other's frames to make sure we're talking about the same universe. But TRF is "prerelativistic": the subtext of virtually every post is, "if anyone else prefers a different frame than me, it can only be because he's an imbecile, or a resentful failed student, or an enemy of science."
Furthermore, to call this attitude "the typical arrogance of string theorists" would be unfair to other string theorists. Lenny Susskind, Raphael Bousso, Juan Maldacena, Scott Shenker, Ori Ganor, Michael Douglas, etc. have all been incredibly patient at various times in explaining physics to me, and I'm grateful to them. But I've also had the experience of having them ask meand not (it didn't seem) out of politenessto explain the halting problem, or P versus NP, or pseudorandom generators, or quantum faulttolerance, or even "physics" issues having to do with (say) entanglement theory or the relation between bosons and matrix permanents. Watching these "masters of continuous math" struggle to understand what, for me, were freshmanlevel points only added to my admiration for them. When confronted by something alien to their own training and mental style, their attitude  being the excellent scientists they were  was to try to understand it.
Lubos, you're 39 years old. Isn't it time you asked yourself whether Susskind, Maldacena, and the others  many of whose contributions to string theory exceed your own  might have been onto something here?
Now, did talking to these guys humble me about how much physics I still need to learn? Of course it did. But it ALSO cured me of any inferiority complex I might have felt about "my kind of math" being trivial child'splay to a string theorist, or boring and irrelevant to those struggling to understand Nature. The empirical data is in, and it isn't. :)
Contrary to what you say, I have no intention of forcing physics into a narrow mathematical straitjacket. HEP theorists should continue to use whatever mathematical tools they need to make progress  even nonrigorous or borderlinenonsensical tools! On the other hand, if they later (as many have) become curious about the sorts of questions at the physics / information theory interface that their kind of math can't address, but my kind of math can, then people like me, and books like Quantum Computing Since Democritus, will be standing by ready to help. :)
Personally, I love the way Roger Penrose once put it: Minds are just a tiny part of the physical universe. Math is just a tiny part of what minds think about. But physics is just a tiny part of what math can describe. So one could say that math, physics, and computer/information/cognitive science each subsumes and "trivializes" the other two in turn!
One correction to what I wrote: Scott Shenker > Steve Shenker. (Steve is the string theorist; Scott is his brother the computer scientist. :) )
Dear Scott, I have outgrown these pissing contests while I was still in the kindergarten. After all, I don't have any personally driven preference because I know *all* the disciplines you mentioned. ;)
However, it's still true that theoretical physics is the most fundamental layer of knowledge about Nature  I mean the real world  and every attempt to replace physics research by a completely different discipline and its methods is bound to end up with gibberish. Everyone with a basic understanding of science knows it.
The other disciplines are more fundamental in studying whatever they study  but what they study is not fundamental among all other topics! Note that some other TRF readers mentioned that Amy Farrah Fowler's claim that neuroscience "incorporated" physics and was therefore superior was just silly. It's great to hear you confirm that you are indeed your own clown cartoon and you believe this idiocy yourself.
I always highly appreciate it when reading about mathematical I am not too familiar with, such as the things explained in chapter 1 and 3, when their importance and application in phyisics is applied too. So thanks for doing this here Lumo :)
Roger Penrose links the first mathematical chapters in "The road to reality" nicely to cool physics too. That is why I quite like the first half of this book and more or less strongly discagree with things he writes in the later chapters ... ;)
Obviously it depends on which questions you care about! If what you want to know is the specific Hamiltonian of the universe, then knowing that the universe has the computational power of BQP would help you only very slightly with that (by ruling out some crazy possibilities). At some point you need to go out and build a particle accelerator. I don't think I've ever met anyone stupid enough to dispute that.
On the other hand, the converse is also true: if what you want to know is whether the universe can be efficiently simulated on a digital computer, then all the knowledge in the world about the details of our universe's Hamiltonian won't help you much with THAT.
Now, these are both excellent questions  and personally, I don't see any principled reason why we ought to regard the first as more "fundamental" than the second. The fact that the first kind of question has been studied since the time of Galileo and Newton, while the second kind of question is historically more recent, is not a good argument.
You are agreeing with my remark, I think. I was expressing that one can never make any correct statement about Nature that does not incorporate both indeterminacy and superposition. These cannot be pictured or described classically. Turing machines it seems. Those classic pictures with determinate known past measurements and with a discrete output certainly in a definite state. One can paint a picture with dots but it will never be accurate on closer inspection.
Hi Eugene! I do work as computational linguist  contractor to companies and universities. I am personally fascinated by how semantics changes over time in scientific language. The science is more fundamental than the language meaning, though, the language just reflects the changing expanding knowledge underneath.
Not all mathematicians are obsessed with issues irrelevant to physics. The reason I want to know what kind of mathematical axioms are needed to do physics properly is that those are axioms I can trust to actually be true, so that theorems I prove from them (even ones that have nothing to do with physics) gain epistemological soundness.
It is possible to reject the axiom of choice and have all sets of reals be Lebesgue measurable (although to make calculus work it is convenient to use a much weaker form of the axiom of choice that can't give you nonmeasurable sets, but which lets you make a countable sequence of choices depending on earlier ones, so that all the forms of convergence you need are equivalent). But that alternative version of mathematics is deprecated by mathematicians, even though there is no real difference as far as concrete statements are concerned.
In particular, any statement of Number Theory (or statement provably equivalent to a statement of Number Theory, or statement of analysis without too many quantifiers, which covers almost all the important open questions) can be proven from ZFC iff it can be proven from ZF. The Choice Axiom simplifies many proofs a great deal but it is only strictly necessary for very abstract results like the BanachTarski paradox. Therefore the resistance to assuming that all sets are measurable is sort of religious; no amount of physical evidence can prove or disprove it. (On the other hand, sufficiently concrete mathematical statements can be proven or disproven with high confidence by physical evidence.)
Another way of putting this is "mathematics is not logically prior to physics". (Well, I'll grant that some very concrete mathematics may be logically prior to physics...). I agree with Kurt Godel that eventually we are likely to modify our fundamental mathematical axiom systems in the light of their usefulness for physics. But I wish physicists would try harder to explain to mathematicians what axioms work better than ZFC for proving things they want to prove.
I can put this more concretely. The axiom system which takes ZF without the Axiom of Choice but adds the axiom that all sets of reals are Lebesgue measurable is a valid alternative to ZFC, and it even proves that ZFC is consistent which is something ZFC itself can't do. In this alternative system, would it be easier to make Quantum Field Theory mathematically rigorous?
When you said "the other examples are different but morally similar  that physics leads one to prefer other axiomatic systems than what a person unexposed to physics, experiments, and reasoning about them could a priori think dogmatically", what other examples of axioms did you have in mind?
Dear Joe, I have some doubts whether lowlevel choices such as measurable subsets of real are relevant for the actual problems that make it somewhat hard to define QFTs rigorously.
I would guess the answer to your question is Yes, this system would make it easier to define QFTs rigorously. However, what you really need to deal with in QFTs aren't just sets of reals but functionals and perhaps sets of functionals and one encounters all kinds of genuine, tangible problems with renormalization issues etc., not just the "liar says he is a liar" type of binary naive problems that set theorists spend 99% of their time with.
The questions from the two scientific disciplines  and the relevant axioms and theorems  are much further from each other than you seem too assume.
I wasn't suggesting any axiomatic QFT here, which I don't consider an important thing, anyway, when I was talking about the preference for measurable sets. It was more about a natural axiomatic framework for more primitive layers of physics.
Hi Lubos. I disagree with your statement that computers have nothing much do in undemantal physics research. Contemporary computers can't do any creative work like we humans do, but on the other hand they can process and store massive amount of data which we humans can't. The sky surveys we are conducting with the help of computers are good exapmles. There's no doubt the we'll be able to maximize our understanding of dark matter and cosmological parametres through these surveys. At least for that matter, Physics will depend on Computer Science. It's not just Physics, all subjects in one or other way depend/will depend on CS. And there's is no SHAME in saying that.
Also personally, I felt your arguements comes from a deep rooted personal believes that "Physics is above everything". You are a highly judging personality, aren't you? Yes, of course everyone judges, but people like you judges very quick. You prefer to approach things with a presumption rather being open minded. I think that's the reason why deptite of your intellectual abilities, you haven't achieved much as a theoretical Physicists.
I'd like quote Heraclitus here: "Wisdon is one thing: to know the thought by which all things are directed through all things"
Dear thinkerbof, our exchange with Scott wasn't about whether computers are useful tools for those who are doing fundamental physics. One could perhaps defend the "yes" answer  just like good food, chalks, and blackboards may be useful. (Although whether the examples of the astro activities you mentioned should be called "fundamental" is debatable, too.)
Instead, the topic of the exchange was whether computer science  a part of maths about the ways how computers work  is fundamental for the understanding of how Nature works. That's the question that I answer with a No. You are trying to distort which question was being discussed which is why you fight a straw man.
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