The unreasonable effectiveness of mathematics in the natural sciences (1960) [pdf] | Hacker News Reader

The unreasonable effectiveness of mathematics in the natural sciences (1960) [pdf]

maths.ed.ac.uk

113 pointstruth_seeker70 comments

I 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.

Math 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.

Example of physics' 'unreasonable simplicity'?

S = k. log W

Everything but gravity seems exquisitely described, if you look close enough, by 22 parameters.

Could you elaborate about the 22 parameters? Are you referring to universal constants or fundamental particles?

There 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?".

Physics 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.

>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).

> 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

But is there a better way to understand things?

Some of that larger scale complexity may be due to irreducable emergent relationships that reductionism alone can't explain without essentially simulating those reductionist relationships to the scale you need from the ground up (which defeats some of the goals of reductionism).

You may need perfect information in order to fully simulate such systems (a snapshot of all initial conditions and perfect knowledge of physical laws). This may be the case for many scales of relationships we're interested in.

Current reductionist theories can't tell me if I'm going to have coffee tomorrow morning. I don't expect them to either, yet they still have very useful purposes. They may never be able to tell me if I have coffee tomorrow and I'm fine with that (maybe due to error propagation inherent in quantum mechanical principles or the need for say emergent models).

To 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

It 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.

>> 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.

This is a very straightforward consequence of using zero volume point masses.

In 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_.

The blowup shown by Xia doesn't rely on point particles. It works just the same even for "small spheres". I didn't really make that clear.

Your thought is in good company though. Lots of physicists over the years have worked to remove point particles from the classical theories. This issue is that these fixes tend to end up fixing nothing (like with Newtonian gravity), or producing nonsense and contradictions elsewhere. A famous result in electragnetism, if we replace point charges with tiny charged spheres, we get solutions where particles accelerate exponentially without any external forces. Not at all obvious, but what the hell, right?

Anyway, to take a card from your deck and muse haphazardly, my guess is that these problems are extremely hard to avoid when dealing with a continuum. There is some interesting work on discrete theories, but I'm not all that informed about them.

The problem with points (which is also present in electrodynamics) is a wonderful example of when the hole in your present theory points the way to the next theory. We already know that Newtonian mechanics does not predict the behavior of the very small, so it is fitting that it doesn't pretend to give an answer. You could almost suggest that it points the way to the diffuseness of quantum mechanics.

The limited, and hence reasonable, effectiveness of mathematics in physics

http://arxiv.org/abs/1506.03733

> 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"?

When 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

For 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.

Maxwell's equations seem unreasonably simple. But their successor, electroweak interaction, is far more complex. Who knows what complexity an electrostrong theory might hold?

Electroweak 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.

The 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.

The philosophical question is how much math describes the universe and how much it describes the things our mind considers understandable.

To 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.

2017: https://news.ycombinator.com/item?id=13954804

2014: https://news.ycombinator.com/item?id=8520610

I'm surprised those are the only two given it's from the 1960s. There really needs to be a section for timeless content like this.

We have that! 'Hacker News Classics' http://jsomers.net/hn/

Amazing, I hadn't seen that.

The 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?

"Is there any example of something that isn't considered math but describes nature?"

- "The apple is red"

No math, describes nature.

Inevitably 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.

The Aplle reflects light with a peak at a wavelength of around 600 nm.

Wow. 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?

For 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.

> experiment analysis

And without that it isn't science.

> The 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?

No, "math" and "descriptions" are the same concept. As soon as you describe anything, you're using math.

Why learning about things by examining what they're like is considered "unreasonably effective" is more of a mystery.

Is 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?

I 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.

> not everything can be explained with math

Can you give me any examples? I'm curious.

infinity

Was 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.

there is no math for dreams

Couldnt it be modeled as electrical activity inside specific areas of the brain under certajn circumstances?

It's the same as: There are more real numbers than you can write down. (Everything you can write down using symbols can only be enumeratable many (you can enumerate all symbol combinations starting with single symbols), but there are uncountable many real numbers)

So now you say: Could you give me an example of such a real number that we can not write down?

Answer: No such example can be given because even e or pi or any other real number constructed by an equation is given by just inventing new (countable many) symbols (like π) or by writing down an algorithm which consists of countable many symbols.

So to answer your question: There could be problems that can not be explained by math but in the same way you might not even be able express/name/describe the problem.

If framed as a logical equation, wouldnt math explain or have an answer?

Sure, i may not write an answer, when thinking from an algebraic perspective, but i sure can answer it from a logical or sets perspective.

Perhaps the answer is an empty set?

I am not sure I understand what you are saying. If you are speaking of equations you are speaking of symbols right? (∧,∨,¬,...) Whose combinations are only countable many.

The answert to what is the empty set?

>Could you give me an example of such a real number that we can not write down?

Thst this is a question without any answer can be quantified as a logical equation. The question might not have an answer that is a number, but has a superseding answer thst it is not logically consistent.

Well the answer to: > Could you give me an example of such a real number that we can not write down?

Is clearly: No, I can not.

But the answer to the question "Are there real numbers that can not be written down in _any_ way" is clearly: yes. If the question is understood as "Is the set of reals bigger than the set of naturals?".

So there are cases in which the answer to "can you give me x?" != "is there an x?".

So back to you original question:

> > not everything can be explained with math > Can you give me any examples? I'm curious.

I guess nobody might be able to give you any examples but they could still exist.

The same is true for functions. There exist many more functions (eg ℕ→ℕ) than you can write down because the powerset of ℕ is bigger than ℕ. So it could be that there could be a physical law that the property X of an object is a function of property Y but that function is one which we could not even see as a function because we can not encode it in symbols and probably our brain would not even recognize it as function at all (?)

I’m not sure exactly what you mean by your challenge, but there are several standard examples of noncomputable numbers.

because even e or pi or any other real number constructed by an equation

Not true. Most real numbers cannot be constructed at all [1] [2]. There is some interest in replacing the mathematics of real numbers with computable numbers, but this adds additional complexity to analysis (requiring a computable modulus of convergence).

[1] https://en.wikipedia.org/wiki/Computable_number

[2] https://en.wikipedia.org/wiki/Specker_sequence

Yes that's what I was trying to say with the later part of my setence you quoted:

Only coutable many real numbers can be given by name (like pi) or by an algorithm. But there are still many more numbers left which you can not name/point to/give as example.

I think this is an example: Math works well for electronic circuits, but it doesn't work well for psychology. Maybe coincidentally, we can predict circuits but not people.

I'm surprised by the downvoting on this thread. Kumarvr asked for examples of things math doesn't explain. Simple examples of things math doesn't (at this time) explain (e.g. psychology; dreams) were downvoted.

Why? There's no way to know why. Just downvotes. Maybe someone thinks math explains physics, which explains [...], which explains dreams. Maybe that's not it at all. Who knows?

I can only speak for myself here, but silly downvotes are the primary reason I'm disengaged with this otherwise really great site.

> "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.

Without myself making any claim about whether maths is or isn't behind everything....

that we may only be able to discover mathematical properties does not itself justify saying that "Math isn't behind everything, and not everything can be explained with math."

There is a large amount of work (I don't know what would be a complete proof for this) behind the idea that anything we can explain, we can do so with math. If we can't explain something with math, we can't explain it, period.

It's just an incredibly flexible tool.

Perhaps that points to our cognition being a mathematical product.

Are there ways of understanding things in a non-mathematical - meaning non-cognitive - way? Is there something greater than Turing Completeness?

Such an idea is silly because we're using reason to fathom the answer.

When we encounter some sort of alien cognition out there which is beyond the bounds of math, it either will be understandable and expand our mathematical vocabulary as we seek to formalize it...

Or it will look just like gibberish and passed off as 'too chaotic, or too self-referential', or some other unfathomable quality.

They are two different questions and both kind of true in my opinion.

You can use math analyse anything and say try running regressions on interest rates but that the reality is different from the equations and made of people doing stuff.

In particle physics you get things matching the equations so accurately that one might suspect that the reality actually is math.

An amazing book I’ve been reading is

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.

Reminds me of this:

https://youtu.be/HEfHFsfGXjs

That's because everything can be approximated with maths. Even with shoewhat random events you can tame it with statistics.

If ever there was a argument for Intelligent Design, this would be it.

Well, 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.

> ...precise laws that conserve both matter and energy; therefore, nothing gets created and nothing gets destroyed...

Not your main point but just wanted to point out that because of inflation, energy actually isn't conserved! It's kind of surprising, but consider light getting redshifted from a far away galaxy. Its frequency drops as it travels through space and thus, so does its energy. This happens with the beam not interacting with anything, so the lost energy doesn't really go anywhere.

Thanks! That is truly fascinating. I have never encountered that idea of it being a consequence of the red-shift before but it does make sense. I wonder where that energy goes. Perhaps there is an ether after all? Seeing as Planck implies that nothing is continuous in this universe, then I wonder what the quanta of the redshift is? Perhaps time itself causes the friction. (I apologize for my layman's spitballing here but this universe is nothing if not fascinating.)

And by inflation, I was referring to the brief period shortly after the BB where the clumps of galaxies quickly separated themselves. Is that what you are referring to here, or are you referring to the general expansion of the universe?

I say the opposite, I see this as a compelling argument AGAINST intelligent design.

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