Article Review ⇢ THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES -Eugene Wigner
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- Опубликовано: 19 май 2024
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"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."
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↳ QM ➜ amzn.to/48Xu8mx
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Science is a phenomenal exploration of nature. We hope to hone our skills of problem solving by exposing ourselves to multiple contexts. In doing so, it can sometimes be challenging to see the connection between topics. I yearn to understand 𝙝𝙤𝙬 these aspects of physics, unite together. To accomplish this, I'll cover all of my old textbooks through QFT; the convergence point of the many modern scientists! These posts are very much in a "𝘯𝘰𝘵𝘦𝘴 𝘵𝘰 𝘴𝘦𝘭𝘧" style. 𝙈𝙮 𝙝𝙤𝙥𝙚 is that by sharing this exploration, I can help others navigate the beautiful world of mathematics & physics through problems and examples, connecting the mathematical tools to their physical ramifications.
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◉ ☞📚📖📓= Griffiths, David J., and Darrell F. Schroeter. “Chapter 4 Quantum Mechanics in Three Dimensions.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 131-197.
◉ ☞ 🖼 📸 = tinyurl.com/4v9nef5k
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As many of you have pointed out, or have messaged me on Instagram, there is another paper that rebuttals this article. The review of that paper, "The Reasonable Ineffectiveness of Mathematics" by Derek Abbott, is now live!
ruclips.net/video/ovLdbAUjrFM/видео.htmlsi=iVpPw9EdeQJ7KjUM
The real reason for the uncommon effectiveness of mathematics in the natural physical sciences is because the most fundamental of all of the accidental modes of being is Quantity.
I don't think I have seen this perspective before!
But that's actually the point of it.
First we observe qualitatively that two properties seem to be related to each other under specific circumstances. Or we suspect them to be.
In the first case we start measuring and based on the results we start modelling a mutual dependency of the properties with the circumstances described as a system. The description as a system is just as important because it provides for two other important aspects:
reproducibility
transferability or rescalability
But exactly those two aspects are depending crucially on the aspect of quantity and measureability.
In the second case we allow a framework of known quantitative relations between properties to suggest a new quantitative relation which inspires experiments to assess the correctness of our assumptions about a new rule.
With the advent of classical physics according to Galileo, Kepler, Newton, Leibnitz, Huygens, etc. the applicability of mathematical in physics became established as a universal principle turning quantitative description into an essential criterion of validity.
The discovery of the discontinuous character of some properties in nature and the relevance of statistics as a fundamental concept in nature have promoted the relation of mathematics and physics to a new level of insight and abstraction.
@@michaelburggraf2822 Well put! measurability is also a huge problem in modern experiments. Nice point!
I think people forget that a lot of mathematics was developed for the purpose of explaining the world (newton, calculus, and gravity for example). Hence it is no surprise at its usefulness. It would be like asking why a hammer is useful for building a house. Well it’s obviously useful since that’s what its intended purpose was.
Of course, however, many advanced subjects do not really have applications, nor are they studied for that reason (number theory, algebraic geometry, set theory).
Lastly, I will say that what is means to do mathematics is very ill defined. For you can’t say it’s “the study of numbers” as many mathematicians don’t study numbers (they may study shapes or sets or something). Thus, mathematics is really just a way of thinking. It’s a framework for solving problems: first set some definitions, then layout some assumptions, prove some theorems, and then give some examples. This is the common structure among all mathematics books. So, I don’t see the fuss about why it’s useful as its exact purpose was to be a tool for solving problems, whether real world problems or problems just for fun (pure mathematics).
This is a common answer and it fully misses the point. Yes, there were obviously areas of math that were developed out of contact with real world phenomena and/or to solve some specific problems: like basic arithmetic or geometry.
But how did it turned out for example that Lie Groups and Lie Algebras turned out to be the basis upon which Quantum mechanics is built? It’s not like arithmetic and geometry. It’s a case when a son of human mind has been born not with a purpose to help quantum mechanists, but it just turned out to be helpful. How to explain it? Luck? Is it not surprising at all that some pretty abstract math developed not to solve specific problems turned out to be able to solve these problems? I don’t know. But the answer “it’s like being surprised that hammer can nail” is definitely based on wrong understanding of how some mathematical objects gain their applicability
@@timothytiberius487 Yes I agree with you example on Lie algebras being developed “purely”, that is, without the goal of applying it to the natural world. Two things:
I think we need to remind ourselves that theories of physics are incomplete. And that word “incomplete” means “incorrect” in the logical world. It means that one of our assumptions is wrong. For example, the assumption that wave functions exist in quantum mechanics. We could be way off. So I don’t think we should immediately grant mathematics a win since it “explains” small particles. For should we grant Mathematics a win because it “explained” gravity in the Newtonian sense? No, because Newtons’s theories are wrong: a wrong assumption about the mathematical model behind gravity.
Secondly, if mathematics is effective in describing the natural world, I would attribute its power to its abstraction and generality of being a way of thinking, as I mentioned above. When I learned linear algebra, my professor explained a group to us using a set of ducks, with an operation of reproduction among the ducks that produces another duck in the set. Here we’ve used group theory to explain ducks, but I would hardly say this “explains” the natural world.
Using both of the above, what I am trying to say is that when physicists are looking for some tool to develop their theory, they use whatever is best at the time. For newton, it was calculus. For QM it’s linear algebra. But such assumptions that physical structures are indeed explained by these mathematical objects could be completely wrong and therefore mathematics should not be considered unreasonably effective.
A lost wanderer in the woods stumbles upon a path and it leads him out. Does he say “Man, it’s so unreasonably that this path was effective!” The path could’ve been built for pure exploration or it could’ve been built by someone prior who also needed a way out
, well, couple of points. First, when we recognize that our physical theory based on some mathematical framework is incomplete, we usually just find a better theory with other mathematical framework. Whether this process will be infinite or not is a big different question on which we can only speculate. But even if we were always wrong, ok, we can go with a more humble thesis: math unreasonably provides best approximations and if anywhere we are close to the claim “we really know smth about nature’s organization”, this is where we use a rather abstract mathematical apparatus. And it feels like this thesis still needs some attention.
As for your second point: you used the word “explanation”. And I think it’s not necessary to mention explanations here. Applicability is a wider notion than explanatory power. For example, predictable power is big part of applicability. And as I heard from physicists, machinery used for QM provides us with perhaps the most accurate theory(currently) in terms of match between experimental data with the calculated data. And is your second point kinda says that mathematical machinery used in fundamental science is kinda a totally unnecessary strained analogy(like you example with ducks and groups)? Or that most probably smth else will be fundamentally more effective and accurate? If this what it was about, then it sounds like rather strong statements and only time will show if they are correct.
I personally don’t know what to respond about the “unreasonable effectiveness…”, but I feel that lots of people very quickly come to a conclusion that “no, the entire presupposition is wrong, there is definitely no unreasonable effectiveness…”-and I think they are too quick here; it’s a more complicated and non-obvious matter
@@timothytiberius487 yes I agree, there are two things to decouple, but both should be addressed. One is whether or not mathematics is effective (hence my points about incomplete theories) and the other is, if mathematics is effective, then why. As I said, I don’t think we should even claim point 1. But suppose we do, I guess I just don’t see why it’s not obvious. Why does a spatula kill a fly when it was designed for something completely different? Who knows, it just works, and I don’t think there’s some “hidden meaning” behind mathematics I guess is what I’m trying to say
Thanks for the discussion . As a mathematician I like hearing what philosophers have to say about what it is that we do
@@BennettAustin7, I’m not a philosopher. I’m a mathematician with interests in philosophy
You should read the rebuttal.
Abbott, Derek. "The reasonable ineffectiveness of mathematics [point of view]." Proceedings of the IEEE 101.10 (2013): 2147-2153.
I am already half way through this! Thank you so much for the recommendation! I am going to have to cover this article in a video too. Awesome perspective with respect to engineering!
@@curiousaboutscience Looking forward to a video on that one.
@@uploadJ It was such a fun read! Should have it ready soon!
Thank you again for this suggestion! I finally made the time to record the review of this rebuttal - ruclips.net/video/ovLdbAUjrFM/видео.htmlsi=iVpPw9EdeQJ7KjUM
@@uploadJ Should be live now! ruclips.net/video/ovLdbAUjrFM/видео.htmlsi=iVpPw9EdeQJ7KjUM
Once upon a time there was a fruit-fly named Wiggy whose brain, like his eyes, was composed of hexagonal pixels. So it always saw the Universe in terms of hexagons. One day Wiggy took a PhD in Physics at the local human university, and wrote a paper about the amazing effectiveness of hexamath, that forecasts the entire behavior of the universe. But his thesis advisor said, that's a foolish paper. The universe is not forecastable by hexamath, but by features-math, aka as categories math, that is: the universe can be forecasted via variables, because the human cortex converts all signals into categories (via the Vernon Mountcastle algorithm), to which it then gives names, and represent by ink (or chalk) squiggles, which we humans manipulate, to forecast the universe's behavior. Well, said Wiggy, that's what I do. But, said his thesis advisor, don't you see that you can only grasp the hexa part of the universe? Well, said Wiggy, what about you? You can only grasp the categorizable parts of the universe. Well, what other parts are there? said the professor. I can't tell you, said Wiggy, because your brain doesn't have hexa-pixels. Then to try to find some middle ground, both try to study Quantum Mechanics, where there are no categories (until the probability function goes pfffft) and no hexa-pixels either, but were stumped, until an Alien EBE came down in a flying saucer and said he could explain it all, but the explanation used neither hexamath nor categories math, but something else based on EBE brain, so he didn't, and the problem stayed unresolved. Or did it?
I was not ready for such a response, but I am glad you commented! I can't remember the podcast, but there was a similar argument presented. What a fascinating demonstration regarding the concepts of the nature of perception, understanding, and the tools we use to interpret the universe. Seems as though there will always be a philosophical question: To what extent is our understanding of the universe limited by the cognitive tools at our disposal?
Perhaps the universe might always have aspects that elude our understanding, no matter how advanced our theories or diverse our perspectives.
@@curiousaboutscience Nah. Too many long words. Try shorter ones: The universe is a bunch of shmoo, which we can "perceive" only by putting the shmoo on a grid of attributes / categories / features invented and named by our cortex. The universe has nothing to do with either the features or their names. The "collapse of the wavefunction" is really our cortex choosing this or that other feature to handle the shmoo. The problem arises below the Planck scale, where there is only shmoo, no features. So you can have the proton both "greater than" and "smaller than" its parts. Of course, it is neither of these, but also both. Because categories do not exist in the shmoo below a certain size. At least above that size your cortex can invent them. But at a lower scale it cannot. That's the fun of Quantum Physics. You can understand it only if you are a fruit fly. (But you can never tell your understanding to a human, alas.)
@@thesleuthinvestor2251 however physics and mathematics are providing means of at least partially testing the reasonability of a description of nature.
Due to its abstraction character at least parts of a different description should be possible to be reformulated in a mathematical way so that that part could be checked for equivalence with a different formal description.
In physics approximations should allow for testing if a description is consistent with a different theory or hypothesis.
Examples for both comprises classical physics as an approximately solution of quantum mechanics and the equivalence of the theories of Heisenberg and Schrödinger or the equivalence of the QED description of Feynman and Schwinger.
@@thesleuthinvestor2251a Freitag fly may seem small to you but it's still orders of magnitude bigger than the typical scale of quantum processes and particles (except cooperative phenomena).
The thing went fwoosh is definitely the most accurate depiction of the event.
🤯
Galileo believed that nature was inherently mathematical, that mathematics was the language of nature-that mathematics was the key to understanding the reality behind the appearance of natural phenomena (for example, accelerated and parabolic motions).
Do you believe Galileo's stance is correct?
@@curiousaboutscience
Mathematics is a language.
Physicists speak it, chemists speak it, biologists...
Without this language you come no way.
Why does science can be best dedcribes by mathematics?
Perhaps the creator of the physical world created first the spiritual world. Mathematics is spiritual.
Yes I think Galileo Galilei was correct.
Please do share the article link. Makes life a 100 times easier for those who want to study further.
Did you check the description
@@noproof7376 It only shows pdf. "Paper link" or "link to the article" ud be more useful. And yes, on top of making life 100 times better, it will make life 120 times easier, since I won't have to type such comment.
@aniksamiurrahman6365 you can find it in 5 seconds by just searching the title of the paper. it's the first result.
IIt's in the description as pointed out, or easily found online. However, I can change the name to help viewers find the paper more easily-that's a simple fix. Thanks for the feedback! Happy learning! 🙂
Isn't this a bit like wondering why paints are so unreasonably effective at representing the world?
Math gives us the ability to picture the world.
So does painting.
Potentially, but then you could say what about photography? Or any other medium in which we choose to describe the world. Some might be better than others, and questioning why they are or are not is interesting.
I think it's weird to say that. It's like saying language is a bad tool too. Besides, our models sometimes stay wrong for centuries. Is it that effective?
I suppose we would have to consider the scale of time in the argument of effectiveness. As a tangent though - is an incomplete theory better than not hypothesizing a theory? Assuming wrong is looked at as bad - would that prohibit the discussion of potential theories/models for the sake of correctness. I am not sure what to think about this yet, but I think wrong models at least can give insight to potentially more accurate models that wouldn't have been thought about otherwise. And in this sense, they might be viewed as effective.
Mathematics is the science of solving problems. One branch of this science is the development of abstract models, and the development of methods for solving problems which arise. Met with a problem in the physical world, a master of the subject can usually find a suitable model where the methods have already been developed.
Seems like this could be a good framework to build from. With this definition of mathematics, it really easy to see that abstracting the basic principles out to their boundaries allows for several options when trying to match physical observation with models.
@@curiousaboutscience I've long held this view. If you consider any science, they all have different branches. Consider just the field of notation. Scientists develop notations for particular investigations, and so do mathematicians. Some notations are replaced by more useful ones. For example, Newtonian dot notation for derivatives is rarely used now. And the original notation for chemistry is long gone.
Furthermore, the subject should be taught as a science in schools - we might develop more, and better, mathematicians if that were done.
I disagree.
By mathematics you can solve problems, however, mathematics is much more,
Games like Chess, Checkers, Go...
Philosophie
Logic, ...
In mathematics proofs exist.
Outside of mathematics we usually have no proofs.
Perhaps we can say:
Everything that is provable is mathematics.
@@bernhardbauer5301 No. Mathematics does not guarantee solutions. Study Godel.
@@rchas1023
Have I said mathematics garanty solutions?
Why should I study Gödel?
I am a little confused.
Math is just the study of patterns. The fact that it applies so effectively to the natural sciences means there are patterns in the universe.
19:40 Cosmology in a nutshell
Do you study cosmology? If so, what are some big problems, I suppose philosophically speaking, that you have noticed?
@curiousaboutscience
I don't study it. Simply follow the latest notions. I'm 75 and enjoy watching how the Intersection of clever and duh chases its tail. As for physics and such, I follow [SH] on her channel. I'm more of a mathematician
@@christopherellis2663 That's a nice way to put it, "clever and duh chases its tail". In light of the new video, do you follow the Platonist view of mathematics?
Is just a language ... a very objective language for human sized 2 bodies movements, and not much more that this, certainly not the episteme
This sounds on par with what the rebuttal paper is starting off with. I can't wait to dig into the paper fully and upload another video covering this concept!
@curiousaboutscience I used to have a very "mathaphisical" ontological almost pitagorean view of math, as the base nature or something. Now, I think it is a really well developed language description of motion as it appears in human perception, I think the extremely rigorous discourse of math is false premise, and anything close to it is merely derived from our very biological needs of counting time, and food, and people, and recognizing motion, I mean, math describes the logic and limits of our perception, I think that is more a structure of our perception then some fundamental fact about reality. Can't wait for the next video!
@@ericalves5514 If I am interpreting this response right, you are highlighting the concept of anthropocentrism in the development of mathematics. In which case, the rebuttal paper definitely highlights this aspect. Here is the video, ruclips.net/video/ovLdbAUjrFM/видео.htmlsi=wfN0kOtU40TNRVyp
Would love to hear your comments on it too!
Just a few comments: I think the central idea of mathematics is symmetry. Even somethin g very simple like an equation: x=1; this is a symmetry that means that variable x and the number 1 are equivalent and can be substituted, in some kind of formulation using the variable. This symmetry gets stretched over enormous intellectual distances in mathematics, such as the notion that pi is somehow related to statistics in a Gaussian distribution. The pi in the Gaussian distribution is the same as the pi in a circle - its a symmetry that is not immediately obvious but true none the less. The question of whether some kind of alternate formulation exists that could explain other phenomena that is different from current mathematics is interesting, but it would still depend on symmetry in some other "direction". If we discard symmetry entirely and just look for some kind of coincidence for explanations of natural phenomena, then we are no longer using science. The reason that symmetry can explain so much is that it reduces the complexity of an explanation.
Thanks for sharing! I really like this framework too, and I believe the likes of 3b1b have helped to show the, not so obvious, connection in this symmetry example. I suppose a meta question would be if there is a symmetry to, say, parallel formalisms. I can think of one with matrix mechanics and wave mechanics in quantum mechanics, but the comment above by @thesleuthinvestor2251 is making me curious about others.
Symmetry is NOT THE central idea of Mathematics. Just A central idea.
@@samueldeandrade8535 What would you say it "THE" idea in mathematics, if one exist.
@@curiousaboutscience such question doesn't really make sense. It is the same as asking
"What's your favorite thing of all things?"
It is too general. Making a choice is to drastically limit the universe set. Etc, etc, etc.
So, to answer you question: I would never give an answer to that. It simply doesn't make sense. Not to me in particular, not in general.
Goedel?
bro you sound like moist critical
Thanks? haha
Hate it when ppl being unreasonable
Mathematics is the language of the universe as much as English is the language of God.
Pixels, paint, letters or numbers, the medium used to represent reality is just that: a medium of representation.
Cope more
English is no language just an amalgamation of many things
So is this medium more effective than others?
@@curiousaboutscience Is it effective? Perhaps. Is it reality? Perhaps not.
That old tale of "did we create or discover mathematics" is over. We could not have created such a masterpiece 2,500 years ago, when we first used it. The immutable laws of Math, Algebra, Arithmetic and Geometry are discoveries, each exposing the others to us. We cannot have one without all the others. With them all, we measure and describe the Universe.
Does seem to feel that way! A few principles abstracted to their boundaries. Either way, glad to be able to learn about them all!
ABSOLUTELY WRONG! All the advances in the foundation of mathematics and mathematical logic made 100 years ago (!) mean we now know and understand what maths is and how its works. The idea of 'discovery' in maths is an illusion. I like to use the metaphor of CHESS. A chess player may rightly feel that they have 'discovered' a new chess strategy, or play, but in fact every possible game of chess that could ever be played is already implicitly contained within the rules. We as people are exploring and elaborating what is already there for our own benefit. However, no one would argue that the game of Chess was discovered rather than invented. It is NOT part of nature, it is a human artifice invented for human benefit. So it is with maths. People needed to count sheep, so they invented natural numbers. People needed to keep their financial accounts, so they invented integers. People needed to share out land, so they invented rational numbers. People needed to measure how much water there was in a tank, or the mass of a sack of flour, so they invented real numbers. People needed to solve certain types of equations. so they invented complex numbers and so on. All of maths is simply a matter of choice. It is based on one set of axioms (assumptions) which prove useful because they have the richness to generate the mathematical structures we have found useful in the past. No one can say that there aren't other sets of axioms which might also lead to a rich diversity of different mathematical structures we do not currently use, but which might prove useful in some future human activity. There might be any number of alternative mathematical universes we could invent if they are useful, like inventing a different game to chess, like 'Go' or something. An intelligent alien civilisation, which lives in a completely different environment to us, and has evolved differently, might solve their problems using a different maths, it just depends what structures it produces, how richly it produces such structures, and how they match the problems the aliens have. Maths relates to the human mind, nothing else. It is a form of quantitative language.
Already the title is silly.
There is nothing unreasonable in mathematics.
Even so the equation 2+2=4 is correct up to 777 digits this is perfectly reasonable.
Would you say there is anything unreasonable then?
@@curiousaboutscience
Yes.
The believe that something material comes from nothing is unreasonable.
The believe in dark matter is unreasonable.
The believe in dark energy is unreasonable. By the way, what is energy?
The believe that plants have developed in/from primordial soup is unreasonable.
Up to now nobody has created such a soup. All chemical elements are available, all temperature ranges are available, however, we do not even have a computer program for such a soup. Has anyone created a cell from scratch?
The idea that mathematics is 'unreasonably effective' in the natural sciences is just silly. It is a fetish of people who still, despite all the advances in mathematical logic and the foundations of maths in the 20th C, believe in some parallel 'Platonic universe' where mathematical objects exist. Mathematics is completely defined and completely explicit, or it would NOT be mathematics. It is a precisely defined language invented by people to measure things quantitatively. To say that mathematics is 'unreasonably effective', is the same as saying the English language (or any other similar) is 'unreasonably effective' at describing the world. Anyway I think it highly debatable that mathematics is 'unreasonably effective' in the natural sciences just as English has limitations in describing the world, which is why we invented maths in the first place, maths itself has similar limitations. Engineers who have to apply maths to messy complex situations in real life are more likely to use numerical methods, semi-empirical approximations, statistical approximations, and any other number of 'fudges' to get results. Mathematical models trying to describe reality are prone to break down such as using Navier Stokes to describe fluid flowing around a 90 degree bend. This world view of purists is a delusion which should have been dispelled 100 years ago, once we understood what mathematics is and how it works.
This is very in line with the rebuttal paper. I am currently reading through it, and you are right on with the approximation and numerical methods being using to get something "useful" out of whatever the context is.
@@curiousaboutscience Thanks
Wigner was a pretty good physicist, but a laughably bad philosopher.
I wonder how he would write this now - given the current state of physics and philosophy.