Saturday, July 19, 2014

What is a theory, What is a model?

During my first semester I coincidentally found out that the guy who often sat next to me, one of the better students, believed the Earth was only 15,000 years old. Once on the topic, he produced stacks of colorful leaflets which featured lots of names, decorated by academic titles, claiming that scientific evidence supports the scripture. I laughed at him, initially thinking he was joking, but he turned out to be dead serious and I was clearly going to roast in hell until future eternity.

If it hadn’t been for that strange encounter, I would summarily dismiss the US debates about creationism as a bizarre cultural reaction to lack of intellectual stimulation. But seeing that indoctrination can survive a physics and math education, and knowing the amount of time one can waste using reason against belief, I have a lot of sympathy for the fight of my US colleagues.

One of the main educational efforts I have seen is to explain what the word “theory” means to scientists. We are told that a “theory” isn’t just any odd story that somebody made up and told to his 12 friends, but that scientists use the word “theory” to mean an empirically well-established framework to describe observations.

That’s nice, but unfortunately not true. Maybe that is how scientist should use the word “theory”, but language doesn’t follow definitions: Cashews aren’t nuts, avocados aren’t vegetables, black isn’t a color. And a theory sometimes isn’t a theory.

The word “theory” has a common root with “theater” and originally seems to have meant “contemplation” or generally a “way to look at something,” which is quite close to the use of the word in today’s common language. Scientists adopted the word, but not in any regular way. It’s not like we vote on what gets called a theory and what doesn’t. So I’ll not attempt to give you a definition that nobody uses in practice, but just try an explanation that I think comes close to practice.

Physicists use the word theory for a well worked-out framework to describe the real world. The theory is basically a map between a model, that is a simplified stand-in for a real-world system, and reality. In physics, models are mathematical, and the theory is the dictionary to translate mathematical structures into observable quantities.

Exactly what counts as “well worked-out” is somewhat subjective, but as I said one doesn’t start with the definition. Instead, a framework that gets adapted by a big part of the community slowly lives up to deserve the title of a “theory”. Most importantly that means that the theory has to fulfil the scientific standards of the field. If something is called a theory it basically means scientists trust its quality.

One should not confuse the theory with the model. The model is what actually describes whatever part of the world you want to study by help of your theory.

General Relativity for example is a theory. It does not in and by itself describe anything we observe. For this, we have to first make several assumptions for symmetries and matter content to then arrive at model, the metric that describes space-time, from which observables can be calculated. Quantum field theory, to use another example, is a general calculation tool. To use it to describe the real world, you first have to specify what type of particles you have and what symmetries, and what process you want to look at; this gives you for example the standard model of particle physics. Quantum mechanics is a theory that doesn’t carry the name theory. A concrete model would for example be that of the Hydrogen atom, and so on. String theory has been such a convincing framework for so many that it has risen to the status of a “theory” without there being any empirical evidence.

A model doesn't necessarily have to be about describing the real world. To get a better understanding of a theory, it is often helpful to examine very simplified models even though one knows these do not describe reality. Such models are called “toy-models”. Examples are e.g. neutrino oscillations with only two flavors (even though we know there are at least three), gravity in 2 spatial dimensions (even though we know there are at least three), and the φ4 theory - where we reach the limits of my language theory, because according to what I said previously it should be a φ4 model (it falls into the domain of quantum field theory).

Phenomenological models (the things I work with) are models explicitly constructed to describe a certain property or observation (the “phenomenon”). They often use a theory that is known not to be fundamental. One never talks about phenomenological theories because the whole point of doing phenomenology is the model that makes contact to the real world. A phenomenological model serves usually one of two purposes: It is either a preliminary description of existing data or a preliminary prediction for not-yet existing data, both with the purpose to lead the way to a fully-fledged theory.

One does not necessarily need a model together with the theory to make predictions. Some theories have consequences that are true for all models and are said to be “model-independent”. Though if one wants to test them experimentally, one has to use a concrete model again. Tests of violations of Bell’s inequality maybe be an example. Entanglement is a general property of quantum mechanics, straight from the axioms of the theory, yet to test it in a certain setting one has to specify a model again. The existence of extra-dimensions in string theory may serve as another example of a model-independent prediction.

One doesn’t have to tell this to physicists, but the value of having a model defined in the language of mathematics is that one uses calculation, logical conclusions, to arrive at numerical values for observables (typically dependent on some parameters) from the basic assumptions of the model. Ie, it’s a way to limit the risk of fooling oneself and get lost in verbal acrobatics. I recently read an interesting and occasionally amusing essay from a mathematician-turned-biologist who tries to explain his colleagues what’s the point of constructing models:
“Any mathematical model, no matter how complicated, consists of a set of assumptions, from whichj are deduced a set of conclusions. The technical machinery specific to each flavor of model is concerned with deducing the latter from the former. This deduction comes with a guarantee, which, unlike other guarantees, can never be invalidated. Provided the model is correct, if you accept its assumptions, you must as a matter of logic also accept its conclusions.”
Well said.

After I realized the guy next to me in physics class wasn’t joking about his creationist beliefs, he went to length explaining that carbon-dating is a conspiracy. I went to length making sure to henceforth place my butt safely far away from him. It is beyond me how one can study a natural science and still interpret the Bible literally. Though I have a theory about this…


  1. On your closing "wonder", an alternate theory: whether religious or not, we systematically overestimate the order-independence of learning. Of course the end result information of learning a sequence of exact facts is logically independent of the order in which the facts are learned; but it seems real learning in the wild is not all about exact learning of exact facts.

  2. All my mathematical models are incomplete.

    Gödels' theorems only hold (apply) for complete models asserted as such.

    Pretty clever, huh?

    That way I don't have deductions that lie outside the scope of what the model says is true or false.

    "This deduction comes with a guarantee, which, unlike other guarantees, can never be invalidated. Provided the model is correct, if you accept its assumptions, you must as a matter of logic also accept its conclusions."

    True. There conclusions that can not be invalidated...or validated.

    The guarantee still holds.

    Now what?


  3. True. There [are]conclusions that can not be invalidated...or validated.

    Typo (an omission.) see brackets []... above for the correction of the post above.

  4. Hi Bee : Interesting article but I have to say that all religions are not against science. I do not know if you are familiar with eastern religions or not. But these religions (Hinduism and Buddhism) do not have any problems whatsoever with science. So if you had a co student from India or from any number of Asian countries, you would not have any problem!! As a shameless personal promotion, I can perhaps mention my relevant guest blog on a physics website you may not care about. So I will not name the blog web site but interested readers can find the blog easily by googling. Title: Hinduism-for-physicists! Thanks.

  5. I think it is explicitly not clear now if Godel's ideas are to be considered theory or models in this more modern sense.

    As far as the mathematical properties and observations the nature of meta-mathematical proofs depend on the uniqueness and general understanding of prime numbers as the general case.

    Yet somehow the labeling of indistinguishable things in order makes it possible to solve some equations just beyond the models. Such questions of order seem at once independent and dependent on the context of information theory.

  6. kashyap,

    But as I understand it Buddhism was a scientific response to the multiplicity and diversity of the many gods of Hinduism... a philosophy that does not address a religious nature of wisdom or the Buddha. So it seems at once a unity of view as well an internal struggle within these stances toward Western materialism and abstract division of landscapes into Cartesian grids.
    Is it an artifact of three dimensions that Hinduism and Christianity have a trinity, say of a compromise idea of quarks?

  7. Bee,
    I think you make many good points that clarify general rules for doing physics, and science in general. But as so often happens, you backtrack in this post from some previous positions you've taken and do not come to the obvious conclusions.

    The models, which are the reference picture of the world you are dealing with, are mapped mathematically in a theory. But the only time a theory deserves that nomenclature is when it is mathematically consistent when come at from other parts of the model. In other words it hangs together from very many viewpoints of that model. In reference to physics, that model much be the real world how it looks and acts. If that model is not referencing the real world, or it references the real world in only a very few limited perspectives, then it does not qualify as a physics theory. It may qualify as a mathematics theory but again it must be consistent from many different viewpoints within the model being used.

    Using that as a guide I don't see String Theory qualifying as either a physics or a mathematical "theory". Do you disagree?

  8. I should add that "many different viewpoints" is a bit of a heuristic notion but to a person without a previous agenda usually doesn't present a problem.

    For instance the ads/cft correspondence must remain a conjecture and not a theory in physics, because anti-deSitter space is not the world we live in.

  9. I usually hear 6000 years, sometimes 10,000. That is the first time I have heard 15,000. It must be from a new model.

    Although it was a bit unpleasant for you, I am glad you had the encounter so you can empathize with us in the USA, but I think you would find it hard to believe how bad it is. There are three colleges all within about 80 kilometers from me which teach this stuff, and are accredited by my state (New York) to grant degrees which qualify their graduates to become high-school teachers, such as biology teachers and physics teachers. I know of these three because some of my nephews choose among them when deciding where to go to college. I worked in the state of Ohio for a few years and there was another such college ("Nazarene College") in the town where I worked. Some of my nephews and nieces in California went to a similar college there, "Biola". They meet like-minded people in such places, get married, and raise like-minded children.

    There are many worse problems in the world of course, but this is depressing.

  10. " It is beyond me how one can study a natural science and still interpret the Bible literally. Though I have a theory about this…" The model is 1984's "doublethink" - the ability to hold two contradictory beliefs in one's mind simultaneously, and accept both of them. Obamacare healthcare insurance costs will shrink as healthcare costs explosively grow.

    Dark matter is a detection disaster, arXiv:1407.1052, 1310.0828, 1310.8214, 1306.5534, 1306.3983 Add arXiv:1407.3146 eight days ago. Its "particles" are Wesley Crusher hallucinatory. Milgorm acceleration is a one value universal fit. It can be empirically validated on a bench top within 24 hours in two ways, or in an Eötvös balance within 90 days. Do you believe what experts tell you, or what your own lying eyes (don't) see? No ansatz here, faith!

  11. Creationists have a strong argument though namely that Evolution is just a theory after all. How can you argue against that? The power of this statement is overwhelming.

    BTW Sabine, in your opinion the SM is a model or a theory? David Gross quite often advocates that it should be called the ‘Standard Theory’ and not the Standard Model, I'm not sure why though.

    Finally you say:

    “and the theory is the dictionary to translate mathematical structures into observable quantities”.

    But in the figure it is the other way around. You go from “reality” (i.e. observation) to a model via a theory.

  12. /* Provided the model is correct, if you accept its assumptions, you must as a matter of logic also accept its conclusions.”*/

    Provided... This doesn't tell very much about validity criterions of models, which are ipso-facto equivalent to those of theories in this way. No model can be therefore proven correct with certainty in the same way, like the theory and vice-versa: until the postulates of theory remain correct, you should accept its conclusions as well.

    The common mistake of stringy/loopy models is the mistake in logics: they're using correct postulates, but they're confusing intrinsic and extrinsic perspectives for example, because the theorists don't have the geometry of particular manifold before eyes. We already discussed this issue here in connection to entropic balance models at the event horizon of Schwarzschild metric. The consequence for example is, the SUSY theorists predict the existence of their particles at the opposite side of mass energy spectrum, despite their derivations are otherwise quite perfect from formal perspective.

  13. kashyap,

    Yes, I agree with you, religion doesn't a priori have to be in conflict with science. This isn't a post against or even primarily about religion. If I write about religion, or its collisions with science, I am careful to clarify that I mean monotheistic religions in their literal interpretation.

  14. Eric,

    You write "But the only time a theory deserves that nomenclature is when..."

    As I said in my post, it's entirely irrelevant what you or I or anybody else thinks when something 'deserves' to be called theory. That's not how language works. I just tried to convey how I think the word is in practice used.

  15. Giotis,

    The standard model is a model. The theory is quantum field theory or yang mills theory, etc.

    About the figure: Yes, sorry about that. The arrows should go both ways. I reused this figure from an older post and noticed to late it doesn't quite fit. I should update this at some point, thanks for pointing out.

  16. The Creationist guy didn't believe in that stuff because he was religious, but because he was convictional. Even scientists can be convictional. Some believe in something for which there's no evidence at all. And some stubbornly cling to that belief despite the patent evidence to the contrary. When you see this sort of behaviour in somebody who thinks they're rational and logical scientist, it's scary. A good example is the speed of light in vacuo. It isn't constant, but many physicists are convinced it is. See this Baez article:

    "Einstein talked about the speed of light changing in his new theory. In his 1920 book "Relativity: the special and general theory" he wrote: "... according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity [Einstein means speed here] of propagation of light varies with position." This difference in speeds is precisely that referred to above by ceiling and floor observers."

  17. @ L.Edgar Otto:
    The reason I wrote that guest blog is that there is lot of misunderstanding about Hinduism in the west. If I answer these religious, philosophical, metaphysical questions here, Bee and many other readers may not like it. So I would request you to go on that blog and ask questions. I will be very glad to answer them there. Thanks.

  18. Don't get me started on young earth creationists! I'll try to stay on topic...

    One of these guys, supposedly a real scientist, once wrote: "With a model you can prove anything you want."
    Of course he was 'arguing' against models, especially computer models, that clearly and objectively demonstrate the possibility of all kinds of emerging phenomena without the need for a creator or imported blueprint information.

    Apart from the silly notion that all models be worthless, he seemed to totally overlook the fact that his wisdom of reality is... a model, residing is his head. A model therefore that is extremely complex, highly inexact, mostly unconscious, and unverifiable by other people.

  19. i can make models with math, and I can also make models using a quantitative tool which is not math, but is Equivalent to math.

    If you would like to see it I can show it to you.

    My claim is that I can solve any mathematical problem using this tool, and the tool itself is not math, but is Equivalent to math.

    Here's a link to it if you would like a look

    My question for you is simply this. If we are using mathematics to model things, we say "x exists", and this is an inherently certain statement. It is not uncertain whatsoever. But we can also say that "x might exist", and if it has a very high potential to exist, say .999..., then these are really the same thing quantitatively because .999... =1.

    Actually I have a LOT of interesting ideas on that FB page you might want to see. Please contact me if you have any questions or criticism, all feedback welcome obviously. Thanks :)

  20. "Yes, I agree with you, religion doesn't a priori have to be in conflict with science. This isn't a post against or even primarily about religion. If I write about religion, or its collisions with science, I am careful to clarify that I mean monotheistic religions in their literal interpretation."

    The "it's just a metaphor" stuff reminds me of the "no true Scotsman" fallacy. In other words, whenever something in religion is found to conflict with science, some deem it to be a metaphor and hence immune to the conflict. However, all of these metaphors were taken literally before science came along. Also, there is no internal algorithm to decide what is a metaphor and what not. Not everything can be a metaphor if the religion is to have any sense at all. Finally, in some cases, one must ask "a metaphor for what?".

  21. ? I don't think I said anything about metaphors. I find some elements of religions are interesting as, let me call it mental exercise.

  22. Not directly, but metaphor is what is often claims when science disproves the literal interpretation.

    I agree about "certain aspects" though; I quite like someone's self-description as a "secular pagan".

  23. Metaphors can only be understood by those who are capable of understanding them. If you are capable, i.e. spiritually advanced, you have no need to ask "metaphor for what?" Here's a great example from Rome:

    In his “Creation of Adam,” Michelangelo places “God,” represented by a common “metaphor” for wisdom at the time, an elderly, bearded man, inside the human brain approximately where the pineal gland would be; many art critics and historians call this the “Uterine Brain” and erroneously interpret it to mean that Michelangelo was suggesting “God” controls humankind via the mind.

    The truth is, Michelangelo was Illuminati, a member of the Brotherhood of the Serpent most probably, and "The Creation of Adam" is esoteric knowledge hidden in plain sight. It refers to the PHYSIOLOGICALLY transforming process one must go through in order to "reverse the fall."

    When Michelangelo painted “The Last Judgment” in the Sistine Chapel, many of the bishops and cardinals were offended at all of the nudity; one cardinal even suggested to the Pope that it was better suited to a bathhouse. The Pope sent a letter via courier to Michelangelo telling him to “make it right.” Michelangelo sent a letter back to the Pope telling the Pope, “make nature right and art will soon follow.”

    Michelangelo’s point was that the problem wasn’t with the art, rather, it was with nature – the bishops and cardinals. The bishops and cardinals weren’t illuminated; they had not yet transformed themselves into Adam; they were still dwelling in the "fallen state." If they were illuminated they would feel no need to hide nature’s beauty behind a cloak of deceit, they would understand the metaphor.

    Apparently the Pope at the time was also a member of the Illuminati because “The Last Judgment” remained as Michelangelo painted it until after Michelangelo died and even then there were some mysterious and rather humorous difficulties experienced during its defacement.

    Metaphors define a hidden language called the "Twilight Language"; it's used to convey secret information that only the "initiated" can understand. It's only the uninitiated who see problems between religion and science and there's just as many in the ranks of science as there are in the ranks of religion!

    But yeah, you’re absolutely right about the Church. Recently Karen King, the first woman to chair the School of Theology at Harvard and an expert on Coptic Greek and papyrus scrolls, was given a new, previously unexamined scroll in which Jesus refers to his wife ( Naturally this caused a bit of consternation, especially after it was proven authentic. The official position of the Catholic Church was that it was to be taken “metaphorically!” They’ve never taken anything metaphorically in their entire existence!

    There are very few people who know the truth about the story of Christ; you can bet your last dollar that Michelangelo had a pretty good idea.

    In all honesty though, scientists and mathematicians are guilty of their own sleight of hand tricks. For instance, I once read a rather convincing proof, irrevocably accepted by the community at large, demonstrating that any polynomial of degree n has no more than n distinct roots. It was probably the most convincing bunch of bullshit I’ve read in a while. Here’s a counter-example, a characteristic polynomial from a simple S-matrix:

    - x^3 + x^2 + (5/4)x – (1/4).

    That little polynomial is a true anarchist (antichrist?); it even kicks the foundations of Calculus to the curb. Take the first derivative and see if it defines the slope at every point of the curve. See if you can use it to find all of the relative maxes and mins. Maybe the fundamental theorems of calculus are meant to be metaphorical . . . or maybe they should come with an inherent probability, i.e. they’re right 90% of the time! Ha, Ha, Ha . . .


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