Since I want to understand the world, I learn physics from textbooks. But I feel there is a gap between the textbook and the world. I do not know
why the equations in the textbook can control the whole world? How can I bridge the gap? In research, how can one distinguish whether what we do is meaningful in reality or not?
Any help or suggestions or comments will be appreciated.
The world is a very complicated place, so to understand it physicists use two strategies:
For example, to understand the motion of the solar system to pretty good accuracy you just need Newton's laws of motion and gravity, as learned by generations of schoolchildren. However this ignores relativistic effects, so for example you can't understand the precession of Mercury unless you use a more complicated theory - general relativity.
Alternatively, if you're trying to understand fluid flow you use an approximation that treats the liquid as a continuum and ignores the fact it's made from molecules. Again this works well in most cases, but you can't understand Brownian motion unless you take the effect of molecules into account.
Studying only a limited part means, for example, that if you're designing car suspension you don't have take the gravity of the Moon into account. On the other hand you can't understand the motion of the oceans unless you take the gravity of the Moon into account.
So the notion of some equations that describe the whole world isn't terribly useful. You choose whatever mathematical model is accurate enough for the system you're studying.
Someone is bound to mention string theory as a potential fundamental theory. However even if string theory does fulfill its potential, you still wouldn't use it to calculate fluid flow or planetary dynamics.
I'm more with Dilaton than John Rennie here and I think the answer is kind of obvious, but you should take heed that a much greater thinker than I doesn't think so: see Wigner's famous article.
Now that you've been duly warned: in a nutshell mathematics can be thought of as the language one must naturally speak when trying to describe something objectively and without bias. We humans have many prejudices - our senses and particularly our brains have evolved in very specific conditions - those of the wet savannas of late neogene / early quternary Eastern Africa to recognize and react to and cope with the patterns that we encountered there. These are a very specific and very restricted set of patterns - not very representative of the World as a whole that physics seeks to describe and study. We need some way of rising above the prejudices that such a "sheltered" and "restricted" upbringing hinders us with.
One of the ways we do this is through abstraction in the sense of paring away all extraneous detail from problems we think about in physics. We "experimentally" find that there seems to be a repeatability and reliability in logical thought and description, in paring things back to sets of indivisible concepts (axioms) and then making deductions from them. These two actions of abstraction/ axiomatisation followed by deduction are roughly what mathematics is. Add to this a third action: checking our deductions /foretellings by experiment and you've got physics. Mathematics and physics are not greatly different in many ways. Mathematicians and physicists both seem to baulk and bridle at that thought, but I've never quite understood why. One is the language of the other. If you want to get into Goethe's head, then you'd be a fool not to study German thoroughly - likewise for physics and the need to study mathematics.
Interestingly, the idea of stripping extraneous detail away and seeing what the bare bones of a description leads to - an way of thinking that I'm sure many physicists can relate to - was given a special name and prosecuted most to its extreme to create whole new fields by a mathematician - this was the "Begriffliche Mathematik" (Conceptual Mathematics) of Emily Noether that begat great swathes of modern algebra, beginning with ideal theory in rings. The only difference between this kind of creation and that of physical theories is that we are obliged to check the latter with experiment - so that limits which ways deduction can go.
Many of the axiom systems of mathematics are grounded in very physical ideas, even though it may not be that obvious. Much abstract mathematics is created to "backfill" intuitive ideas about the physical world: a good example here is the theory of distribitions to lay a rigorous foundation for discussing ideas like the Dirac delta function - the idea of a fleetingly short, immensely intense pulse. You can trace even the most abstract and rarified mathematical ideas to ultimate physical world musings. The example I like is that of the mathematical notion of compactness and a compact set - one for which every open cover has a finite open subcover. Surely this one isn't a physical world idea? Actually, this notion was the last of many iterations of attempts to nail down what it was about the real numbers that gave rise to the physically / geometrically intuitive idea of uniform continuity. Various definitions can be seen - notably ones where compact is essentially taken to mean "having the Bolzano-Weierstass property" in the 1930s until the modern idea (proposed in 1935 by Alexandroff & Hopf) was generally settled on in the 1940s. It's a good example about initutive ideas being clarified by logical thought. The whole of real analysis ultimately comes from trying to pin down the intuitive idea of a continuous line with no "gaps" in it. These ideas come from our senses and our intuition for the World around us, but they are not always reliable. Mathematics helps us clarify these intuitions and sift out the reliable from the misleading - again, helps us overcome the prejudices of the wet savanna animal.
Footnote: I would post this question on Philosophy SE - there are some excellent thinkers there who are also mathematicians and physicists. In particular, I would be so bold as to summon up the user @NieldeBeaudrap to take a look - Neil is a researcher in quantum information who writes very thoughtful and interesting posts there.
Another footnote: readers should be warned that I am what many people would call pretty weird insofar that I pretty much subscribe to the Tegmark Mathematical Universe Hypothesis.
The equations in the textbook don't control anything. Rather, people learned about the world and then wrote down what they had discovered in textbooks. Working out whether those equations are any good involves two activities. First, you think about the explanation behind the equation, whether it is consistent, what it implies about which systems the equation applies to and that sort of thing. Second, work out ways to compare the consequences of different sets of equations that purport to describe the same thing and do experiments to test those equations.
Why could people work out those equations? There is a lot we don't understand about that, but there are some ideas that seem relevant. Part of the explanation is related to the universality of computation - the fact that computers can in principle simulate any physical object to any finite accuracy. Then there is the fact that it is possible to start with defective knowledge and improve it by guessing and criticism of those guesses. The best discussions of these issues that I have seen are in "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch.
There is not so much a gap between the physical/mathematical description how nature works as John Rennie hinted at, but between the scale of the most fundamental laws of nature and the effective everyday large scale we observe.
The mathematical description of the up to now known laws of nature at the most fundamental scale (without gravity*) such as for example the standard model in particle physics, works extremely well as for example Prof. Strassler keeps explaining by reporting about the newest results in high energy physics, which nicely confirm the prediction of these mathematical quantum field theories.
However, these detailled interactions at the most fundamental scale are not needed to describe macroscopic everyday-scale effects. To describe the motion of a ball, the behavior of liquids and gases, and of any larg system of many degrees of freedom, so called effective theories or mathematical equations are good enough. In principle, the laws that govern the behavior of a macroscopic system are derivable by neclecting (integrating out) the higher energy/smaller scale degrees of freedom. The mathematical procedure that does this is called renormalization and a nice overview of the concepts of effective theories and renormalization is for example given here.
For physicists it is the right thing to persue such mathematical approaches because it works. By this method, they can describe the current established knowledge we have about how nature works, and make predictions, for extending the knowledge into not yet experimentally investigated scales and regimes.
The purpose of research is exactly to try to find our what goes on in such not too well known corners. About the meaningfulness one can generally say that new idea/extension/theory must first of all reproduce the current established knowledge and they must by no means contradict it.
*The standard model is not complete istself since it does not contain gravity among other things. So there are approaches to derive it as the effective theory obtained by renormalizing even more fundamental theories, but for John Rennie I dont go into this here ... ;-)
I will take a perpendicular tack to the other answers.
You ask:
why the equations in the textbook can control the whole world? How can I bridge the gap? In research, how to distinguish whether what we do is meaningful in reality or not?
It behooves every physicist to contemplate the enormous contribution of Newton
Newton also made seminal contributions to optics and shares credit with Gottfried Leibniz for the invention of the infinitesimal calculus.
It is calculus and further mathematical advances that have allowed us to describe nature ( the world) from the stars to the atoms in a manner that not only describes physical situations but also predicts new ones. The equations do not control the world, they describe the world to a certain level of accuracy.
Research in physics has as a gauge the experiments that validate the mathematical theories. Even one discrepancy with the data/reality invalidates/falsifies a specific model and it has to go back to the drawing board. Thus it is the world/nature which controls/chooses the mathematical formalism, and not the formalism the world. In physics mathematics is a tool to describe nature as it is revealed by experiments.
