Physics, Mathematics & Science

[Tim Thompson]

It is my thesis here that mathematics, both pristine, and in the guise of theoretical physics (or theoretical science), is every bit as revealing of the natural world as is experimental observation.

In the Standard Solar Models thread, I pointed out that Eddington had applied fundamental physics, in a study of the opacity of the solar interior, and by comparing with the observed color, brightness & size of the sun, determined that its interior could not be made mostly of iron (or other heavy elements), and had to be made mostly of hydrogen. Upriver objected, commenting that "He calculated. He did not go to the suns interior.". I responded with "Calculating is every bit as valid as going to the solar center, it is not a valid criticism of any science to say that a "calculation" is in any way weaker than an observation, as long as the calculation is based on principles derived from observation. This is exactly the case for Eddington, who used a calculational tool which, assuming it is properly derived, cannot be wrong, because it is derived from well established fundamental principles." And Upriver responded in turn: "That is just so wrong. It is theory. Not reality."

With all that said, I am going to pick up the gauntlet here, and say that Upriver's response is so wrong.

One of the most fascinating things about the universe is that it is truly mathematical. It was that fascination which attracted me originally to physics. There are numerous examples of surprising & unintuitive realities, that were not discovered by experimental observation, but purely by mathematical research. The Uncertainty Principle in quantum mechanics was not discovered in a laboratory. Rather, it is purely mathematical in origin, actually stemming from Fourier theory. Chaos was discovered by Henri Poincare in 1903, long before it became an observational issue, by purely mathematical study. And Einstein deduced through logic & mathematics, that light would travel an altered path through a gravitational field. All of these things were discovered by mathematical & theoretical reasoning, not by "going there" and looking. I think that one of the weaknesses of many alternative views is that they fail to appreciate the real strength & value of mathematical reasoning.

Mathematics, after all, is really only a special case of symbolic logic. The scientific process of inference from observation is similar to the mathematical process, though not perhaps as formal. Nonetheless, just as the logical trail of deduction in mathematics leads to confident proof, so does the trail of inference from observation lead to an equally confident proof in the empirical sciences.

It is all built upon the single assumption that the laws of physics are equally valid throughout spacetime, and are the same. throughout spacetime. Certainly, astronomical observation makes it a virtual certainity that this assumption is true, as we see no reason to assume that physics as we know it is any different on the distant worlds that we observe.

The physics of opacity works just as well inside the sun, as it does inside a laboratory here on Earth. It is not necessary to go to the center of the sun. We can know, to the extent of that word in an empirical science, what's going on there by applying fundamental physics, mathematical physics, to that environment. After all, the fundamental physics we are talking about is all based on observation. So when we do a theoretical analysis of, say, the solar interior, then we are really engaged in the scientific pursuit of inference from observation. If the observations are by definition true, and the inference is logical, then the inference is very likely to be true as well (and in almost all cases experience bears this out).

So I maintain that mathematical (theoretical) reasoning & investigation, are as revealing as experimental observation, and is just as likely to provide valid insight. An over-reliance on observation alone stunts the scientific growth. Theory can be as real as observation.

And what do you think?

[Draconis]

Many times, yes. Real-world mechanisms can be modelled mathematically, and things can just "pop up" that aren't self-evident using observation alone.

Of course, the caveat here is that it can work the other way as well. For years, it was not possible to explain how a Bumblebee flew - yet it did.

[Ken G]

The problem I see with your thesis is that it has two possible interpretations, and which one you are presenting is not clear. The first interpretation seems the more literal, which is that mathematical investigations are in a sense observations of their own. That has only worked a few times, and the problem is, we don't know how many times it has failed and been forgotten. In that stronger form of your claim, it sounds like it can fall victim to the same flaw as Intelligent Design-- if we see complex systems that work really well, there must be something inherently special about them. But we might be missing the huge amount of tinkering and failed tries that came first, so that in the end we only see the finished product and not the messy workbench. Observations take the role of natural selection in this look at the theory process.
But in the weaker form, where your thesis is simply saying that observations alone require support from theory to be meaningful, I completely agree. Without theory to organize and idealize the results of observations, it would all be a hopeless muddle. It is truly amazing that human mathematics really works in this role, but there does seem to be some profound reason why it does. So yes, observations without theory not only have little value, they're not even science. I'm reminded of the joke that a physicist, a philosopher, and a logician were at a conference and looked out the window to see a black squirrel. The physicist says, "that's interesting, the squirrels are black here", the philosopher says "how can you say that? All you know is that particular squirrel is black". To which the logician shook his head disapprovingly and said "no, all you can tell is that that particular side of that particular squirrel is black!" Observations without theoretical support are nothing short of pedantic.

[Jens]

Theory can be as real as observation.
And what do you think?

I would disagree with that last statement. I agree with what you say about the importance of mathematics, though I personally would use "physical laws" as a substitute. I think science is interesting because there are laws that can be discovered and can be applied to know what will happen even if there are no observations. So I do agree that the understanding and application of laws is tremendously important, and in most cases is equal to observations.

But I think that in the final analysis, obseravation has to trump theory. Subject to the caveat that our observations can be flawed for whatever reason. But if theory and observations conflict, I think you have to look at what may have caused problems in the observations, and if nothing can be found, you have to assume the theory is wrong.

Of course, this means theory, not mathematics. Mathematics, like you said, is just a form of logical representation, so by definition I think it can't be wrong. If, for example, we put two things together and the mass is greater than the individual masses, that is not a threat to the mathematical concept that a + a = 2a. Rather, it would make us consider, perhaps, that there is something wrong with the law of the conservation of mass.

[snarkophilus]

Mathematics, after all, is really only a special case of symbolic logic.

I disagree. Symbolic logic is a subset of mathematics, not the other way around. Of course, you can apply that logic in a self-referential way....

So I maintain that mathematical (theoretical) reasoning & investigation, are as revealing as experimental observation, and is just as likely to provide valid insight. An over-reliance on observation alone stunts the scientific growth. Theory can be as real as observation.

The key word there is "can." You have to take both theoretical and observational evidence with a few grains of salt, even if they agree. Sometimes there's just a mistake in the theory somewhere, but sometimes there are flaws in the experiments. As a theoretician, you want your results to agree closely with experiment, but it's okay to suggest that the experiment might be flawed if they don't (this is okay even if you don't know why the experiment might be flawed -- often the person who did the experiment will be able to figure it out, especially if there are no ego issues). But usually the lion's share of the discrepancy arises from making assumptions to simplify the theory.

I hate to say it, but I don't think theory really is as real as observation. There's always a possibility that you didn't consider something and your theory is wrong. When you see something, you usually KNOW it happened. That doesn't mean it's useful, or reproducible, or that you can explain it. But something happened (even if it was just an instrument error). When you have a theory that's gone bad, you do the experiment and nothing happens. That is the difference.

I do agree that some people rely too much on experiment, and not enough on theory. But it goes the other way, too.

[captain swoop]

. For years, it was not possible to explain how a Bumblebee flew - yet it did.

Urban Legend Alert!

[gwiz]

I think this goes back a very long way. AKAIK, it was never formally published, but if you apply the rigid wing aerodynamics of an aircraft to a bumble bee, you find the wings aren't large enough. The aerodynamics of movable wings are a lot more complex, but the general hand-waving argument that bee flight was all to do with vortices on flapping wings was eventually confirmed.

Edit: I've just found this link which gives a possible origin of the story: http://www.sciencenews.org/articles/...1/mathtrek.asp

[Tim Thompson]

I did not mean to imply that I thought theory should "trump" observation. Surely in the empirical sciences, there is no doubt that sound observation of the universe is the ultimate authority on how the universe behaves, at least under the conditions of the observation.

I actually have two things on my mind. One is the profound observation that mathematics works to explain nature. It is not apparent to me that the universe does anything, or behaves in any manner, that is not readily described by mathematics. Since mathematics is an invention of our own, it seems significant to me that it would work so well along those lines.

The other thing is I guess best called theoretical physics, or mathematical physics. In the specific case of the solar interior, for instance, it is an environment which we will (probably) never be able to go to and look at directly. So we diagnose the state of the interior by making inferences, based on observation of the visible surface, but guided by our knowledge of physics in general. I think we can count on this method, properly applied, to give us a trustworthy understanding of what's going on in there. To me, it is every bit as trustworthy as if we had gone & looked directly. I don't see any difference in confidence between the two.

[Ken G]

You can certainly quote the solar neutrino experiment as a very topical source of support for this particular thesis. I think I do see a difference in confidence level, but I agree that straightforward extensions of physics from realms we have observed to other realms where we can think of no reason for the physics to be any different, has a pretty high shooting percentage in science. And I agree there is something profound there, without which I am sure that science itself would be a futile pursuit.

[Nereid]

I have a one, slightly off-centre, comment to add (for now).

Just as the centre of the Sun is inaccessible to 'direct observation', so too is heart of the atom, any atom.

We Homo saps seem to regard 'direct observation' in very visual terms ... if we can't 'see' it, it doesn't exist!

One of the (physics) revolutions of the early 20th century was quantum theory, which (among other things) shook this idea of 'direct observation' like a magnitude 20 earthquake.

Strange to say, an awful lot of folk remain blissfully ignorant of the mind-boggling findings from this revolution ... 'wave-particle duality' pales into insignificance (to me) compared with the resolution to the EPR paradox (Bell's inequality, Aspect, and all that).

So often, I feel that the mysteries of the Sun's core are so banal compared to 'the inner life of the humble proton' (for example), kazillions of which make up most of your body.

[Draconis]

Urban Legend Alert!

Well, until the advent of supercomputers and highly accurate imaging equipment, it couldn't be explained. How did something actually fly with those stubby little wings versus that large body?

You can certainly say "Urban Legend." I first heard this in the early 1970's, from an Engineer. That's a pretty long-duration legend.

[Nowhere Man]

Well, until the advent of supercomputers and highly accurate imaging equipment, it couldn't be explained. How did something actually fly with those stubby little wings versus that large body?

You can certainly say "Urban Legend." I first heard this in the early 1970's, from an Engineer. That's a pretty long-duration legend.

Here is what the Straight Dope has to say about this.

And from someone associated with NASA.

Fred

[Draconis]

Why yes, certainly. But we weren't able to model how it worked until several years ago. IIRC, the secret turned out to be that the bee's wings create small vortices beneath them, increasing lift, and providing the equivalent of an effective wing surface.

Anyways, the point was that it was not possible to model this for quite some time, and to the best of their scientific ability, no one was able to explain how it flew. Not the validity of the argument overall.

*remind me to select a better analogy next time, eh?*

[Gillianren]

You can certainly say "Urban Legend." I first heard this in the early 1970's, from an Engineer. That's a pretty long-duration legend.

Not really. Jan Harold Brunvand, King of Urban Legend Research (well, I think he is, anyway) has traced the origins of some back a thousand years and more, and most back to the 60s or earlier unless they involve something obviously modern, like cell phones, AIDS, and tanning beds.

[StefanR]

Tim Thompson:

One is the profound observation that mathematics works to explain nature. It is not apparent to me that the universe does anything, or behaves in any manner, that is not readily described by mathematics. Since mathematics is an invention of our own, it seems significant to me that it would work so well along those lines.

One is the profound observation that mathematics works to explain nature.Is the mathematics part of nature and thus by exploring mathematics we explore nature or is mathematics not part of nature and by using mathematics we can describe nature?
It is not apparent to me that the universe does anything, or behaves in any manner, that is not readily described by mathematics.
Is mathematics able to describe anything that is not part of nature/the universe?
Since mathematics is an invention of our own, it seems significant to me that it would work so well along those lines.
Did you mean by that, that the human mind invented mathematics and because the human mind is part of nature, mathematics will describe nature accordingly?
Or is it that mathematics is part of nature and our mind is observing mathematics and by that decribing nature?
If we are trying to observe something is the knowledge about the observer equally important as the knowledge of the observed?
Does mathematics describe the human mind?
Can an observation be the same as the observed?
Can mathematics describe itself?
How does pythagorean philosophy relate to mathematics describing nature?

[Weird Dave]

The important point to remember is that every piece of theoretical physics starts with assumptions and simplifications. If those two elements are incorrect, then the resulting theory will be wrong even if every mathematical step is correct.

Relativity is an example of where this worked well: Einstein assumed that the speed of light was constant for all observers, and then derived everything else from that. If his assumption had been wrong, that theory would just be a footnote of history. Ultimately, only experiments will tell whether your assumptions are valid.

[snarkophilus]

Since mathematics is an invention of our own, it seems significant to me that it would work so well along those lines.
Did you mean by that, that the human mind invented mathematics and because the human mind is part of nature, mathematics will describe nature accordingly?

That seems rather glib. One could equally well say that because the concept of gravity is a construct of the human mind, and the human mind is part of nature, we stick to the ground.

Math describes nature because we have invented math for that purpose. Any math that doesn't describe nature isn't used when we're talking about nature. When you're counting photons hitting a detector, you don't use modular 2 arithmetic. You use the full field of integers, because it's the tool we've invented for that job. When you read a floppy disk, however, you use mod 2 arithmetic, because in that case it is the right tool.

You can't always use a particular tool on a system for which it was not originally intended. Our tools are good enough that we often can, and I think that's what he's saying is interesting, but it's not always the case.

It's like having a flat head screwdriver and three types of screws. The flat notch screws are easy to put in with your screwdriver. The ones with the cross notches also go in, even though you don't use the whole notch. Your tool is general enough to deal with both. That's pretty neat. (It really is!) But no matter how you try, you just can't put in the screw with the square head. So you need to go invent a new screwdriver.

Does mathematics describe the human mind?
Can an observation be the same as the observed?
Can mathematics describe itself?
How does pythagorean philosophy relate to mathematics describing nature?

Math can describe the human mind, yes. Some types of neural networks in artificial intellignece research do a very good job of this, showing the same emergent behaviours that people's minds do. (Actually, if we want to be silly about it, we'll define a human mind as follows: Let M: A --> B be a mathematical relation from inputs A to outputs B that emulates my brain. We have a mathematical definition!)

Math can mostly describe itself, too. That's the basis of Gödel's thesis.

I'm not sure what you mean by the other two questions...

[Ken G]

It has been suggested that we revive this thread with the slant of the role off Occam's razor. The idea is, there is clearly some profound reason why mathematics is so useful in physics that it can be argued that doing math is sometimes actually doing physics. Perhaps this profound reason is connected to why Occam's razor works-- i.e., why it is fruitful to seek simple and fundamental explanations rather than tackle the full complexity of the reality around us with a purely empirical approach. The purely empirical approach to science, which I've called elsewhere "google science", by all rights should be the only way that works, and mathematics should not necessarily play a role beyond the arithmetic of keeping track of quantities. But in fact, fundamental principles are very useful, idealizations are fruitful, and mathematics plays a key role. It's all part of the same deep truth that I think we have very little grasp of.

[hhEb09'1]

It has been suggested that we revive this thread with the slant of the role off Occam's razor.

Thanks for bumping this thread. I seem to have missed it the first time around, and I have a few comments.

Perhaps this profound reason is connected to why Occam's razor works--

I would say Occam's Razor works like Astrology works--sometimes it is going to be right, but there is no reliable way to tell when

"Calculating is every bit as valid as going to the solar center, it is not a valid criticism of any science to say that a "calculation" is in any way weaker than an observation, as long as the calculation is based on principles derived from observation. This is exactly the case for Eddington, who used a calculational tool which, assuming it is properly derived, cannot be wrong, because it is derived from well established fundamental principles."

One famous counterexample is Kelvin's calculation of the age of the Earth, which was (erroneously) taken to be a refutation of the ancient age of the Earth displayed in the geologic record.

Many times, yes. Real-world mechanisms can be modelled mathematically, and things can just "pop up" that aren't self-evident using observation alone.

One of my favorite examples of that is the story¹ that Siméon Poisson objected to the wave theory of light because he calculated that there should then be a bright spot in the middle of the shadow of a circular obstacle--which he thought was absurd. Of course, when Arago actually did the experiment, he found the spot.

[1] College Physics, 5th ed., Miller, p.612-613

[Spherical]

Einstein had something useful to say on this subject. I will here quote him from the 1954 (fifth) edition of his The Meaning of Relativity:

The only justification for our concepts and system of concepts is that they serve to represent the complex our experiences; beyond this they have no legitimacy. I am convinced that philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of a priori. For even if should appear that the universe of ideas cannot be deduced from experience by logical means, but is, in a sense, a creation the human mind, without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experience as clothes are of the form of the human body.

Put another way, if you fail to make observations and experimental tests, the Goedel monster will gitcha.

[Ken G]

These are interesting perspectives, and some good experiments and quotes, but what I find most interesting is that Einstein does allow the possibility that "the universe of ideas cannot be deduced from experience by logical means, but is, in a sense, a creation the human mind". If this is true, isn't this by itself enough to validate at least part of Tim Thompson's main point? Yes, thought by itself is of little value, but that's not the issue-- the issue is, why is abstract thought playing a crucial role in science at all? Or if we avow that science is all about converting experience into abstract modes of thinking about experience (i.e., "understanding reality"), then the real question is, why is this possible at all when applied to things outside of the influences under which our brains evolved? There is no easy answer to this question, it may be even harder than the understanding of thought itself.

[Relmuis]

As I understand it, Occams Razor (post 18) is about simple explanations being more often right than complicated ones. The explanation might be statistical.

Suppose some phenomenon P can be explained by a lot of different underlying mechanisms, M1, M2, M3, etc. Only a few of these mechanisms will be simple, while one can think up an endless variety of complex mechanisms.

Suppose that one of these mechanisms is in fact the right explanation. If it is a complex one, it is just one face in a multitude, and any theorist seeking a complicated explanation for P is very unlikely to select it. But if it is a simple one, there may be only a handful of equally simple ones, and any theorist seeking a simple explanation for P has a fair chance (one in five, say) of selecting it.

Therefore, theorists proposing simple explanations have a reasonable chance to be proved right by later tests, while theorists proposing complicated explanations are very unlikely to be proved right.

If this is the true explanation for the efficacy of Occams Razor, there may be a number of phenomena which do have complicated underlying mechanisms, and which will, in all likelyhood, never be satisfactorally explained.

[hhEb09'1]

Therefore, theorists proposing simple explanations have a reasonable chance to be proved right by later tests, while theorists proposing complicated explanations are very unlikely to be proved right.

How would you apply this reasoning to say a real example? Like Newton's theory vs. Einstein's?

[Relmuis]

I suppose one might count the number of parameters, the number of formulae, the highest exponent, the highest differential, and more of such things.

As for Newton vs. Einstein, that wouldn't be the right comparison. Both theories are simple. Newton's theory of gravity, however, is not merely the simplest one which accounts for Kepler's (observational) laws; it is the only one (as Newton himself showed by deriving his Universal law of Gravity from Kepler's Laws). Mercury, the main Solar system witness to the accuracy of Einstein's General Relativity, does not completely conform to Kepler's Laws, so the above statement is not paradoxical. If Keplers Laws are rigourously true, Newton's theory must be rigourously true.

Einstein's theory of gravity, General Relativity, however has its rivals. Theories which might be true, because they do not contradict known phenomena, but which are usually more complicated than General Relativity. Of course, all such theories must obey two constraints: they must conform to Special Relativity rather than Galilean Relativity (as Newton's theory did), and they must approach Newton's theory in the limit for small velocities and small gravitational fields. This limits their possible simplicity, for example throwing out the no-gravity theory that simply says F_1,2 = 0.

But let me give a somewhat silly example. Maxwell's Equations can be derived from Coulomb's Equation and Special Relativity. But perhaps Coulomb's Equation can be replaced by something more complicated. Instead of F_1,2 = k * Q₁ *Q₂ / R _1,2², we might have F_1,2 = k_1,2 * Q₁ *Q₂ / R _1,2², where k_1,2 might be some very complicated function of Q₁ / Q₂ + Q₂ / Q₁. This might yield very complicated alternates to Maxwell's Equations, and yet describe all known phenomena, if k would only differ from its normal value for very, very large differences between the two charges. One day, new observations might force us to adopt such a complicated k, but Occams Razor says that we should not do that until such a day arrives.

[hhEb09'1]

I suppose one might count the number of parameters, the number of formulae, the highest exponent, the highest differential, and more of such things.

Both theories are simple.

So, how did you quantify that?

[Relmuis]

For Newton's theory, there is no need to do that, for reasons stated above. For Einstein's general Relativity, I suggest you apply the recipe to the formula relating the curvature of spacetime to the energy-impulse tensor, and then check the equivalent formula in any rival theory. (The version with the cosmological constant would be one unit more complex than the version without, which may be why Einstein didn't like it.)

In my silly example, the alternate formula would be at least 4 units more complex than Coulomb's formula, but might well be 154 units more complex, if for example some 50th roots are extracted to calculate k.

But I fear that more extensive discussions of what constitutes complexity might detract attention from the actual topic of this thread.

[hhEb09'1]

For Newton's theory, there is no need to do that, for reasons stated above.

I'm not asking, this time, for a comparison of Newton's with Einstein's--I'm just wondering how you determine one is simple and another is not. It doesn't seem straightforward to me, at all.

But I fear that more extensive discussions of what constitutes complexity might detract attention from the actual topic of this thread.

Ken G revived the thread with some intent to discuss Occam's Razor. I'm not sure what he intended by that though.

[Ken G]

It was just the kind of ideas that Relmuis was offering, and the other posts as well, that I revived the thread. These are exactly the interesting issues, why is Occam's razor effective (and amazingly effective, at that). It wouldn't be if physics were like the game of chess, for example, there's really no way to play good chess without just grinding out all the possibilities. Unifying principles and simple ideas are of extremely limited value there, and that's a human creation! What about the universe?

[Relmuis]

Well, I was worried that hhEb09'1 and myself were, between us, starting to highjack this thread, getting it bogged down into uninteresting levels of detail, while the grand ideas would no longer get the attention they deserve, being squeezed out, as it were, by pages of formulae. Perhaps the question What do we mean by complexity/simplicity? deserves a thread of its own. And anyway, my ideas about the way to measure complexity/simplicity are not very deeply thought out: I merely uttered them in response to a question.

I do believe, however, that complexity/simplicity is a valid category, and I suspect (or rather fear) that the efficacy of Occam's Razor may be due to statistical reasons: the number of possible complex theories outnumbering the number of possible simple theories. Like I suggested in post 22. (I hope it is not true, though.)

That said, I will one final time elaborate on my recipe for detecting complexity. It's rather humdrum, I fear. Taking theory T, one extracts those formulae to which it can be reduced, throwing away any formula which can be derived from the other ones plus earlier established theories. Then one counts the number of variables in these formulae, adding the values of all exponents, and the degrees of all differentials. But all these terms must be integers. Therefore, taking the cube root adds 3 units of complexity, not 1/3 unit. In essence, one counts symbols, but with a few restrictions.

The first restriction is that any constant would merely add one unit of complexity, however it is written down. (Therefore the constant k in my Coulomb example is worth one unit of complexity, not five, as would be suggested by substituting 1 divided by 4 times Pi times Epsilon-zero. And the little numbers 1 and 2, which are there to identify the charges, do not add any complexity at all.)

The second restriction is that vectors and tensors do not add an amount of complexity proportional to their number of components. (Therefore the R sqared in my Coulomb example adds only three units of complexity, not nine.) Otherwise, General Relativity might be seen as unconscionably complex, with tensors numbering 16 and even 256 components. But you can't blame a theory for the number of dimensions it is supposed to work for.

However, it might be argued that a vector is still more complicated than a scalar, and a tensor of high rank more complicated than a tensor of low rank. So I would amend my original idea by counting every index as an additional unit of complexity. This would give the R squared in my Coulomb example four units of complexity rather than three -- one would have to write R_mR^m rather than R². It would also add an extra unit of complexity for F itself. And it would give a 4-tensor of rank 4 five units of complexity, rather than just one or a whopping 256.

[hhEb09'1]

I do believe, however, that complexity/simplicity is a valid category,

It is, and I have seen a lot of different attempts at quantifying such concepts. None have been shown to "work", to my knowledge.

Otherwise, General Relativity might be seen as unconscionably complex, with tensors numbering 16 and even 256 components. But you can't blame a theory for the number of dimensions it is supposed to work for.

There's part of the problem--most attempts founder when they are actually applied to specific examples--and then have to be modified to refit. The result usually has other problems. I doubt we'll be able to devise one here, considering all the previous attempts. Wasn't it Guth who came up with a huge list of them?