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The unreasonable effectiveness of mathematics in the natural sciences (1960) [pdf]
- _bxg1 8y agoReminds me of this: https://youtu.be/HEfHFsfGXjs https://youtu.be/HEfHFsfGXjs
- whatshisface 8y agoMath is as far as we can tell a totally universal building kit for anything we want to set up, all the way from elegant models like Newton's laws to billion-parameter monster neural networks. The surprise is not so much the unreasonable effectiveness of math as it is the unreasonable simplicity of fundamental physics.
- elevenoh 8y agoExample of physics' 'unreasonable simplicity'?
- Kurtose 8y agoS = k. log W
- evanb 8y agoEverything but gravity seems exquisitely described, if you look close enough, by 22 parameters.
- thegabriele 8y agoCould you elaborate about the 22 parameters? Are you referring to universal constants or fundamental particles?
- evanb 8y agoThere are 3 generations of 4 fermions, each has a mass. One sets the scale of the theory, meaning we should express all the dimensionful quantities in terms of that mass. So, that gives 11 parameters. Then there are the mixing angles in the CKM matrix. For three generations, that matrix is a 3x3 unitary matrix = 9 parameters. But, we can absorb arbitrary phases into each quark field, but should add one back for the global U(1) phase for 9-6+1 = 4 parameters, 3 mixing angles and 1 CP-violating phase. We're up to 15. We have 3 forces in the Standard Model, each has a gauge coupling. 18. There's the Higgs vev and mass, 20. QCD admits a marginal topological quantity, θ_QCD. As far as anyone knows it's zero to many digits of precision. Still, it's in principle a free parameter of the Standard Model, so 21. Then there's the Planck mass (or equivalently, G), 22, which is the number I remember. I suppose now that we are certain neutrinos have mass I should include the parameters of the PMNS mixing matrix, if you're willing to count them in the Standard Model (they're not in the Glashow-Weinberg-Salaam work, for example). How many free parameters it contains depends on whether neutrinos are Majorana. If they're Dirac, it's 4 parameters (like the CKM matrix), if they're Majorana there are 2 additional CP-violating phases. So, 26 or 28 continuous parameters in the Cosmic Control Booth according to the Standard Model. There's of course discrete kinds of questions you can ask, like "Why is the gauge group SU(3) x SU(2) x U(1)"; I'm not sure how to count such a choice. I think it's fair to ask, "Here's the physical theory with the matter content set, what are the free parameters?".
- matt4077 8y agoPhysics is the poster boy for reductionism, i. e. the ability to split any situation into its parts, understand those, and then reassemble a complete model that actually describes the world. Hence the jokes about spherical chickens (the approach works, but not for chickens). Physics has also taught us to expect absurdly simple laws as the inner workings of seemingly complex observations. e = mcc would feel like a cruel joke on anyone trying to grapple with how reality works pre-Newton. This mindset, and the so obvious wins of physics in its golden age (ca 1900-1970), from the atom bomb to the moon landing, also set unreasonable expectations for the other sciences. Hence, people expected no less than a cure for ageing from the unraveling of the genetic code and the decryption of the human genome. Yet, as it turns out, biology cannot be reduced in the way that physics can. Genetics is a madman’s den of complexity, nonlinearity, and unpredictability. It’s almost literally what you would expect from a few million years of Perl snippets evolving. The same is true of neuroscience, meteorology, sociology, and, to a lesser degree, chemistry.
- cinquemb 8y ago>Yet, as it turns out, biology cannot be reduced in the way that physics can. Genetics is a madman’s den of complexity, nonlinearity, and unpredictability. It’s almost literally what you would expect from a few million years of Perl snippets evolving. The same is true of neuroscience, meteorology, sociology, and, to a lesser degree, chemistry. I would say that we cannot to sufficient degree today (the 0 to 1) that people are expecting from the endless hype, but the mathematics used to describe/locate/correlate brain activity (LCMV beamforming + DSP + MCMC priors), and speeding of the design process of molten salts are getting a lot better/more accessible over time (relying on mathematics from local spin density approximations + gradient density correction theories + UNIFAC).
- denzil_correa 8y ago> Yet, as it turns out, biology cannot be reduced in the way that physics can Most "systems" we have (including physics) are complex, non-linear and dynamic [0]. Some of them are relatively less complex than the others but they are still complex nonetheless. [0] https://www.art-sciencefactory.com/complexity-map_feb09.html https://www.art-sciencefactory.com/complexity-map_feb09.html
- xelxebar 8y agoTo be fair though, even Newton's equations have flaws from a pure mathematical physics standpoint [0]. The problem is that certain classes of initial conditions produce infinities. The obvious case is for particle collisions, where gravitational attraction grows without bound in finite time. However, it turns out that similar infinities can arise even if particles don't collide (cf. Xia 1995). Infinities from colliding particles aren't seen as such a big problem, because they require impossibly precise initial conditions, i.e. the initial conditions sit on a measure-zero set in configuration space. AFAIK, the jury is out on whether we can say the same about non-colliding singularities. It seems reasonable to susept that's the case, but if not then Newtoian gravity alone would be demonstrably broken even in the math. [0]:https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture
- dahfizz 8y agoIt seems all of these "problems" arise because you're taking particles to have no volume (point masses). This feels like a garbage in garbage out kind of deal. If your starting assumptions are physically impossible, it's not surprising to get strange results.
- heavenlyblue 8y ago>> It seems all of these "problems" arise because you're taking particles to have no volume (point masses). No, these problems arise, as the parent had said - because those zero-point particles have a non zero probability of occupying the same coordinates if you're simulating their positions using e.g. floating-point variables.
- dahfizz 8y agoThis is a very straightforward consequence of using zero volume point masses.
- Ay_Ay_Ron 8y agoIn your example, floating point numbers are an abstraction for the underlying math which assumes space is continuous. If we're talking about events over a continuous sample space, the probability of two particles being in exactly the same space is identically zero since we can only assign finite probabilities to _intervals_.
- discreteevent 8y agoThe limited, and hence reasonable, effectiveness of mathematics in physics http://arxiv.org/abs/1506.03733 http://arxiv.org/abs/1506.03733
- denzil_correa 8y ago> the unreasonable simplicity of fundamental physics. How do you measure simplicity and how did you come up to the conclusion that fundamental physics was "simple"?
- Sharlin 8y agoWhen physicists (and mathematicians) say a system is simple they usually mean that it exhibits features such as the following: – High degree of symmetry: laws that govern the system are invariant under various transformations – Reducibility: complex entities can be understood as being composed of building blocks governed by a smaller set of rules instead of being irreducible, ”ontologically basic” – Differentiability: the way the system evolves can be modeled using differential equations and other tools of analysis – Low order of approximation: the behavior of the system can be approximated by power series with a small number of terms; higher-order terms converge quickly – Small number of free parameters: the laws of the system do not require a large number of ad-hoc parameters with arbitrary values
- Sharlin 8y agoFor a completely rigorous definition of simplicity, we can introduce the Kolmogorov complexity of an object, defined as the length of the shortest computer program (in some agreed-on encoding) that produces the object as output when run.
- chalst 8y agoMaxwell's equations seem unreasonably simple. But their successor, electroweak interaction, is far more complex. Who knows what complexity an electrostrong theory might hold?
- frutiger 8y agoElectroweak theory does describe more phenomena though. By analogy, Coulomb’s law is simpler than Maxwell’s equations, and describes fewer phenomena. Also, incidentally, Maxwell’s original formulation was very complicated.
- chalst 8y agoThe claim made before is that the description of physics becomes simpler as you go deeper (and deeper means more general). But that seems to have gone into reverse now.
- Iv 8y agoThe philosophical question is how much math describes the universe and how much it describes the things our mind considers understandable.
- adelsk 8y agoTo be fair, physics is only simple, symmetric and nice when considered in the very high or very low energy spectrum. Unfortunately, in the middle physics gets very hard, and all the cool math tools become unusable.
- dang 8y ago2017: https://news.ycombinator.com/item?id=13954804 https://news.ycombinator.com/item?id=13954804 2014: https://news.ycombinator.com/item?id=8520610 https://news.ycombinator.com/item?id=8520610
- lordnacho 8y agoThe question I'd ask is if you found a description of nature, is there any chance it would be called not-math yet still describe the phenomenon in question? Is there any example of something that isn't considered math but describes nature? Isn't there a tendency for math to grow with science too?
- Conjoiner 8y ago"Is there any example of something that isn't considered math but describes nature?" - "The apple is red" No math, describes nature.
- lordnacho 8y agoInevitably you'll become more interested in what exactly red means and you'll end up with something recognisable as math. If you're not already there.
- maxnoe 8y agoThe Aplle reflects light with a peak at a wavelength of around 600 nm.
- jayalpha 8y agoWow. Interesting statement. But "is" means Existence. To exist there must be likely an observer and time. I think your statement is more complex than you and I thought at the first look. Interesting. And what is "red"? A wavelength? A feeling? A neuron firing?
- air7 8y agoFor example, phenomena relating to humans exist in a different realm. For example statements like "Kids like to play games", "The placebo effect works" etc. Math can be used here for experiment analysis, but not to describe the phenomenon or even define the used terms.
- lordnacho 8y ago> experiment analysis And without that it isn't science.
- air7 8y agoIf ever there was a argument for Intelligent Design, this would be it.
- OneWordSoln 8y agoWell, we have a universe of, perhaps, two trillion galaxies that live together in a regularly nonuniform background radiation that can be predicted by a mathematical model predicated upon our advanced understanding of physics and a single point-source event we call the Big Bang, whose physics are beyond our physical reality's description and require some serious hoops to be jumped through (e.g. inflation) to align its progression with the state of our universe today, and yet we have the model and it accurately predicts the precision of our measurements of it (the CMBR) and even its non-uniformity. The universe has a precisely-defined conversion between matter and energy (thanks, Einstein) as well as precise laws that conserve both matter and energy; therefore, nothing gets created and nothing gets destroyed, only transformed, and yet some people deny that something literally "extra-ordinary" is the Source of it all, stuff and its interrelating laws. To say there is no "Prime Mover" is the height of hubris, for we live in a universe with laws that bely its own creation. For we creatures -- no matter how cognitively well-endowed we are -- to say that we are capable of understanding everything about the Source of this creation is not only absurd, but unscientific and logically unsound. 5/6ths of our universe's matter is missing. The energy that keeps the universe expanding is missing. We have no explanation for how the universe implements gravity and, thus, how it works with respect to the other three fundamental forces. And yet "scientists" claim there is no Creator. The interrelation of all the dynamical systems at work in the various levels of the universe is so precise that even a small variance of one of them -- e.g. the Fine Structure Constant -- would not just break the entire structure; it would render it unmanifestible. No, we can not understand how this universe was created, because, as the time-dependent creatures we are, the Creator of all that exists, including time itself, is literally beyond our comprehension. Knowing that "the Unknowable" exists should be common knowledge by now, especially for the physics community, as they work to transform subsets of Unknowns into Knowns. All you who downvote such opinions think yourselves better than Einstein who himself understood that there was a Creator, even as a non-religious person who failed to crack all the mysteries of this wonderful creation. The problem of hubris is a purely personal problem that involves a completely different mathematics fought in a different realm. Only by accepting the truth of reality can one successfully explore it, my dear friends. Plugging one's ears and yelling, "Blah! Blah! Blah!" is not the way of progress, even if it is the way of social media's self-selecting herds. This grand experiment has shown -- among other things -- that people who can not refute any points of an argument would simply rather make the argument go away. Everyone who knows the history of scientific advancement knows what became of that vast majority who ridiculed Boltzman and Einstein.
- usgroup 8y agoI think so much maths comes from thinking about science; really self evident with the development of calculus, statistics, LA, formal logics, etc. It’s not like there is maths and there is science and it’s shocking that maths applies to science because the two came up independently: they are confounded by each other.
- victor106 8y agoAn amazing book I’ve been reading is https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Universe/dp/1328879984/ https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Univ... The author explains the history of calculus leading up to Newton’s treatise on Calculus and how important calculus is to modern life. It’s behind everything you use from cell phones to containing HIV. Highly recommended.
- kumarvvr 8y agoIs the question "Is math behind everything or is that anything can be explained with math?" a valid question? Even for "chaos" we have a chaos theory. Where we cannot apply function/formulae to physical phenomenon, math tends to use empirical calculation and theory to study systems. Is is that the universe bends to math or math bends to the universe?
- v64 8y agoI like how Bertrand Russell viewed that: "Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover." Math isn't behind everything, and not everything can be explained with math. But it happens that the things that are mathematical turn out to also be quite useful.
- kumarvvr 8y ago> not everything can be explained with math Can you give me any examples? I'm curious.
- stareatgoats 8y agoinfinity
- norrius 8y agoWas that meant as a joke or am I completely missing the point here? Mathematics provides plenty of tools to work with infinities. In plural, because there are quite literally infinitely many different infinities, which we know thanks to maths.
- mistrial9 8y agothere is no math for dreams
- kumarvvr 8y ago
- choonway 8y agoThat's because everything can be approximated with maths. Even with shoewhat random events you can tame it with statistics.