Computer proof ‘blows up’ centuries-old fluid equations
quantamagazine.org
quantamagazine.org
The incompressible Euler equations model a very strange and unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes.
For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device!
Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do.
1) https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.
Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity.
The article dances around the term:
> At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”
> Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time.
Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity.
As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.
Do these swirls shed energy? Is it considered in these equations that for example friction within the swirls would slow them down (and hence not reach a singularity)?
To me (2), a singularity in a field like this means that one or more of the field values "blows up", i.e. goes to infinity as you run the time variable forward.
But how could this ever happen? The Euler equations model the "conservation" (i.e. constant-ness) of three real physical quantities: mass, momentum, and energy. If these three quantities are finite and constant when you add them up over the whole field, how can any part of it "blow up" into an infinite value?
The answer is that the blow-up must occupy a volume that shrinks as the blow-up grows, so the conserved quantities are still constant. The singularity would be infinitely small in space, and have an infinite value of density or velocity (or both).
The hard question is, are these blow-ups merely artifacts of a particular numerical simulation technique, or are they essential somehow to the incompressible Euler equations themselves? That's what these papers are trying to figure out.
To me, an "essential" (i.e. inherent-in-the-equations) blow-up seems intuitively reasonable because of the acausal nature of the field. When you simulate the incompressible Euler equations, it superficially looks like it's a physical fluid doing physical-fluid things, swirling and flowing around. But in a real fluid, a change in one part of the fluid propagates to the other parts at finite velocity, creating real cause and effect.
An Euler fluid's time evolution is not a phenomenon that ripples forward through time in a normal way. Instead, every point in the fluid responds to every other point simultaneously. If you poke a cube of incompressible Euler fluid with your finger, there is no pressure wave that ripples through it, where the fluid parcels push each other along and get out of each other's way. Instead, the whole cube of fluid somehow instantly adopts a new flow pattern that conserves mass/momentum/energy in response to that finger-poke.
1) Note that velocity is a vector, since it has a direction. This means that in 2D the velocity is two numbers, and in 3D it's three numbers. So technically the 3D incompressible Euler equations have four values at every point: one density, and three velocity components, one each in the x, y, and z directions.
2) I'm a numerical simulation guy, not a mathematician. Real math experts have rigorous definitions of a singularity, e.g. in https://arxiv.org/pdf/2203.17221.pdf "Singularity formation in the incompressible Euler equation in finite and infinite time," Theodore D. Drivas and Tarek M. Elgindi.
I don't get it. If the fluid is incompressible, how can density have a value at every point in space? Isn't it just a constant?
So you know something is up.
Systems would be modeled mathematically using a fluid's individual component values, but we were paid for the real-world laboratory data.
https://youtu.be/2Vrhk5OjBP8 With a relevant illustration at 9:41 at where I think these topics intersect conceptually.
So, a question is, can we improve our ability to make such approximations and know something about the accuracy of the solutions we will get? E.g., for the Euler equations, will that approximation of an "incompressible" fluid ever work in practice and, if so, when and, there, how accurate can/will it be?
Or, what about, hmm, just to be picky and pick something, friction on the side of the tube? What if the tube is not a perfect tube?
A few grains of dirt: What if the liquid is water but, like most real water, has some solids floating around in it? Right, we can say, so there are a few grains of dirt floating around in the water, and they won't matter -- to be picky, that's an approximation, and we are likely correct, but where is an actual math theorem that says we are correct or how correct, i.e., accurate, are we? Right, a few grains of dirt -- we don't much care. But that's practical judgment and not really theorem/proof math.
And similarly for other approximations we get as we throw out, modify, or honor real features?
So, as stated, this is too difficult as a pure/applied math research direction. Okay, ..., then, is there anything at all in that direction that might be not absurdly difficult as a research direction?
Or, to be simplistic, we work hard and get a numerical solution to a boundary value problem. Now someone tweaks the boundary. Can we say that our numerical solution is only tweaked? Or, when can we say that small changes in the problem statement will result in only small changes in the solution? Right, we are into some topology and looking for a case of continuity .... Hmm .... If we had some linearity ...!!! Right, the two pillars of analysis are continuity and linearity ...! But here with Euler we were considering nonlinear partial differential equations!
Again I ask, is there any hope we can do anything for some corresponding math??
Which is why your comment is exceptionally worthwhile. I also know enough about fluid mechanics to both understand and appreciate it.
People often wonder on HN what the point of a STEM degree is (after making money). To me I've had a lifetime of pleasure from understanding how things work. It's so much better than things being mysterious black boxes.
I once asked a date if she wanted to understand how airplanes worked. She said no, that understanding them would make her afraid of flying. For me, it was the opposite. Knowing how the airplanes fly and how it all works made me a much less anxious passenger.
What kinds of problems does it solve to know an answer to this question? Honestly curious, please do not take this as offensive/dismissive.
We already know that the incompressible Euler equations can't be a faithful model, for reasons I've mentioned elsewhere in the thread. But I think the hope is that if they can answer these questions for incompressible Euler, then they can eventually extend their techniques to more complex fluid equations like Navier-Stokes, which people generally assume (but can't yet prove) is physically reasonable.
Simulation has great practical value, but it doesn't give you any guarantees about the behavior of the solutions for all the cases you haven't actually tried.
If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it.
If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics).
Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying.
1) https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation.
For example, if you're simulating charged particles moving around, and you use a force equation F = k q1 q2 / d^2 (1), then when d approaches 0 (i.e. when the distance between particles approaches zero), then the force F goes to infinity.
For atoms, it works the same way. If you use a force law like Lennard-Jones (2), it also has the interatomic distance in the denominator, so the equation has a singularity baked right in.
You could always adopt a more complex force equation that doesn't have a singularity in it. But in practice, it's easier to use a simple but singular equation, and then selectively ignore its bad behavior.
Here's a cool expository article about blow-ups in classical mechanics and elsewhere: https://arxiv.org/abs/1609.01421
1) "Struggles with the Continuum" by John Baez
https://en.wikipedia.org/wiki/Sonoluminescence
I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.
There’s a real causality bound of the speed of light
Compressible materials tend two have two different characteristic speeds, one for sound waves and one for shock waves.
The speed of sound basically works out to speed = sqrt(stiffness / density). So as a material gets stiffer, the speed of sound goes up. An infinitely stiff (i.e. incompressible) material by implication would have an infinite speed of sound, though this can't happen in any real material.
Shock waves travel faster than sound, at a speed related to the pressure difference across the shock. The greater the pressure difference, the faster the shock travels. So if you had an infinite pressure difference, you could have an infinitely fast shock wave, but again this can't happen in the real world.
However, sound and shock speeds only apply to pressure waves in a material. Other influences like gravity and electromagnetism travel at the speed of light. So for example, if you're doing fluid dynamics for plasma, then you'll have a third characteristic speed, the speed of light, because of the charged nature of the material attracting and repelling itself.
There are also more exotic characteristics, like the speed of a propagating combustion front in a flammable material. But when you get to this level you're no longer just solving one simple set of differential equations.
Mathematicians have split opinions about computer assisted proof. On one hand, there doesn't seem to be a real difference between one page of checkable inequalities vs 500 pages of checkable inequalities. On other other hand, computer assisted proof do not help humans gain clarity on the logical process. The fear is that the computer result will just be an "one and done" result, which people believe is true, but cannot build upon because they don't fully understand it.
OK, maybe it doesn't help people build on the proof, but does it help computers build on the proof? Why does it have to come back to human understanding? Maybe it's not possible right now for computers to generalize proof strategies, but I wouldn't bet against it becoming possible down the road.
In general the word 'model' implies a mathematical construct that mimics the behavior of a real-world experimental or observational system, i.e the experimental or observational data can be generated by the model. Where's the real-world ideal fluid flow? No such systems exist.
> "In principle, if you know the location and velocity of each particle in a fluid, the Euler equations should be able to predict how the fluid will evolve for all time. But mathematicians want to know if that’s actually the case."
At some scale in a real fluid, quantum effects become important, and the notion that it's possible to know both position and momentum falls apart. To quote Peter Atkins:
> "The trouble is when you're dealing with operators, is it turns out you can't always extract explicit information about all of them simultaneously, and this leads to Heisenberg's uncertainty principle, which most people think of as a great confuser of the world and denier of information. I like to think of it as a great clarifier because old-fashioned people, like Newton and Einstein, and Lagrange, and all the people who developed classical mechanics, took it as certain that to specify the state of a particle, you had to specify where it was and how fast it was going."
That's why there are no perfect ideal fluids. Atkins expands:
> "What quantum theory did, through the uncertainty principle, was to clarify - what it said, was discuss the world if you like in terms of positions, or if you prefer, discuss the world in terms of linear momenta, don't try both at once. You get very simple descriptions in terms of positions, you get very simple descriptions in terms of linear momenta, it's only when you - like trying to start a sentence in English and ending in Latin or something, trying to mix the two together, that you get into confusion. So think of the Heisenberg principle as a clarifier - try to think one way, or try to think the other way, but don't try to think like Newton thought."
Why not?
Is it impossible to calculate infinite values in general? I suspect not, My understanding is that a lot of calculus is in fact on how to calculate infinite values.
And a computer is a universal machine, this means that while it can not calculate everything, it can calculate anything that is calculable.
They don't explain this very well, I think.
Take the ordinary differential equation x'(t) = x^2(t), with initial condition x(0)=1. It has the solution 1/(1-t) which blows up to infinity when t tends to 1.
If you try to solve it numerically, using, let's say Euler's method, then this is how you go about it. You pick a step size, let's say 0.1. And iterate this way: you know x(0) = 1, and you also know its derivative x'(0) = x^2(0) = 1. You assume x follows a straight line, so you get x(0.1) = 1 + 0.1 = 1.1. At the next step you add more because x'(0.1) is now 1.21, so x(0.2) = 1.1 + 0.121 = 1.221. You keep going like that.
The numbers will go bigger and bigger, but they will never be infinite. Of course, floating numbers with double precision overflow around 10^308, but if you use a multiple precision library you'll be able to keep going forever and ever.
If you make the time step smaller, the solution will be closer to the actual solution, but still, the algorithm will produce finite values at all times (until it hits overflow).
I believe this article is talking about numerical methods, which are always bound to finite values, because of finite memory.
I would say that it's impossible in a finite universe to calculate infinite values. But it's quite possible to define and manipulate relationships that involve infinity.
You can't calculate the sum of all integers - but you can relate it to the sum of all real numbers. Or compare the sum of all integers lager than one hundred, to the sum of all integers larger than ninety.
A "computer" is just a function from {0,1}^N -> {0,1}^M
Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario.
Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario, which the machine is unable to represent.
If you believe that a computer will eventually be able to accurately simulate a human brain, you might as well give up right now.
Since if a computer with all its constraints is able to simulate a human (brain), but cannot do this, then a human can't do it either.
Conversely if a human can do this but a computer can't, then a computer can never simulate a human.
Don't tell this to a software developer working on AI. They might quit their job and become a baker instead.
I don't think you would have any effect on a mathematician, since they would already be acutely aware some things provably cannot be done.
Computer, as defined abstractly, is just any abstract function from {0,1}^N->{0,1}^M.
Any realisation of that, eg., by providing each {0,1} as an electrical switch, realises physical properties associated with electrical switches only.
The reason that electrical computers are useful has vastly more to do with the electrical part than the computer part. The "computational properties" of the electrical devices we call computers are relatively trivial.
But in any case, no system in virute of being an implementation of a discete function thereby acquires physical properties. A woodern "comptuer" is useless precisely because you can't play video games on it.
Likewise, even if the brain can be described by a discrete function -- (which is so implausible as to be a bit mad and certainly purely an act of faith) -- then it still requires the relevant physical properties to implement. These properties are extremely unlikely to be those of electrical switches.
The "computational work" done by biochemical signalling alone should probably be regarded as "infinite", saying much about the limitations of discrete conceptions of information.
"{0,1}": the set containing the values 0 and 1
"{0,1}^N": a discrete n-dimensional space, where the possible values in each dimension are 0 or 1
So they're saying a computer takes a length N binary sequence input and produces a length M binary output.
(As for "what level is this", I didn't cover any of this in my double A-level[0] in maths/further maths, but I am covering it in brilliant.org and some popular maths books, so my best guess is it's first or second year degree level?)
A computer is an abstract mathematical description (eg., like "prime") of a certain mathematical object, a function.
A computer is a way of specifying a discrete function (ie., one which maps a finite number of bits to a finite number of bits), in terms of a sequence of mathematical transitions.
It's an "algorithmic" way of specifying the domain and codomain of a discrete function.
Electrical digital computers aren't actually computers in this sense, and are extremely aproximately described by them. Inasmuch as the shape of the earth is aproximately "spherical".
In any case, pretty much all of physics does not use discrete functions (indeed, I can't think of a single case). In every way physics describes reality, ie., parameterised on space and time, functions are continuous.
They map an infinite amount of spatio-temporal information to an infinite amount of spatio-temporal information.
And there is yet no reason whatsoever, other than the AI PR machine, to suppose that all of physics is wrong in this regard, and the universe is describable by anything else.
This is relevant here, since the problem that cannot be represented to the machine uses ordinary equations of physics, none of which are computable.
I think the author didn't understand the concepts of precision, range, or of arbitrary precision calculations, and so mades bullshit proclamations without sufficient understanding of the fundamentals.
PS: I notice many people I work with (in MAANG) lack an understanding of Computer Science, aren't formally educated, and lack curiosity. It's discouraging because they also lack concern for code craftsmanship and don't know how to use a debugger. This and they're paid $350-450k / year. If you had to recommend a computer science postgraduate program, which would you choose?
I feel certain that if you run a process that approaches infinity using ordinary floating-point numbers, you will actually reach infinity. This is a case ("can a calculation yield an infinite result?") where computers have less of a problem than people do.
You'd have to deal with the question of whether the infinite value accurately reflected an infinite limit of the process or whether it was spurious. But there's no difficulty in calculating infinite values.
And at any rate, "infinity" is not a single value to be reached.
Take an analysis class and you'll understand this better!
But more likely, the program would have to be built where it understands the symbolic forms involved, more like a proof solver than simple cfd math.
https://en.wikipedia.org/wiki/Quadruple-precision_floating-p...
There are ways of getting floating point infinity which doesn't involve overflow like dividing by there. But with exception of most trivial cases you have the same problem. You can't know whether you had actual 0 or value closer to 0 than what floating point can represent.
All of that of course depends on how floating point unit or calculation environment is configured. It's probably possible to configure it so that overflows/underflows report an error instead of simply returning "inf".
The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity.
And, in general, almost every function isnt representable on the discrete domain above.
floats are a hacky interpretation of discrete bit patterns
> The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity.
This is incoherent nonsense. If you want to say that the floating point value "infinity" isn't really infinity, you must also say that nothing else is really infinity either. That is true in a completely useless and uninformative sense, but it's false in every sense a person would ever use. And it fails to distinguish the correct mathematical proof demonstrating an infinite limit from the simulation suggesting, but not demonstrating, an infinite limit. Neither of them is really infinity. Each of them can represent infinity just as accurately as the other, though.
A paper is just a function from a bounded countable subset of ℝ² to another bounded countable subset of ℝ². What would you conclude, from that, about the limitations of what you can represent on paper?
In any case, no. The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least.
The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity.
A computer cannot demonstrate such a thing, because real-valued functions aren't computable.
No sequence of bit patterns on a computer can ever show the above, because the above is an issue of the limits of a continuous function. Relevant inputs to the function have an infinite precision, and relevant outputs have an infinte precision.
Eg., x = pi, y = pi^100
A computer may very easily mistakenly conclude x = pi is an input which becomes infinite in y, because (1) a computer cannot represent pi; and (2) cannot represent pi^100 either.
infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow"
Why would the output need to be not real? There's no difficulty with saying a real-valued function has a singularity.
The issue is to demonstrate that this function has a singularity at some point, yes. Simulation is a bad way to do that, though conceivably you could get lucky.
> A computer cannot demonstrate such a thing, because real-valued functions aren't computable.
Obviously false; computers are fully capable of providing proofs that some function has an infinite limit somewhere.
> The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least.
That is the only way anything is ever infinity - by designation. As I pointed out elsewhere, IEEE infinity has all the correct mathematical properties of positive infinity in the extended reals, so it's difficult to see what you think you're saying.
> I'm not sure where your misunderstanding comes from, but at least, you might consider you're disagreeing with an article on quanta magazine which writes up a project by experts in their field.
Writing about an expert doesn't make you any smarter. The reason proffered by Quanta is nonsense. They are correct that the experiment they describe cannot achieve the goal sought; they are quite obviously wrong about why.
> infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow"
That's what infinity is. In every sense. Overflowing is defined by exceeding a boundary; infinity is defined by exceeding all boundaries.
I'm morbidly intrigued by your fetish for the idea of "bit patterns". Infinity is also not an image on paper. How do you expect a correct mathematical proof to represent infinity?
This project is about real-valued functions which are taken to describe physical reality. Almost all of physical reality has no closed-form analytical description that "traditional mathematics" can operate on. So there arent any relevant symbolic rules of inference yet invented to resolve this problem.
If you want to program a computer to perform these rules on these functions, there arent any -- hence the millenium problem. And if there were some, we wouldnt bother using a computer.
What "using a computer" here means is finding a discrete approximation to this system, searching through that discrete input space until something which looks "infinity-like" occurs in the output space.
Now, a priori, this is never going to constitute a proof of anything. Since the discrete approximation needs, independently, to be analytically shown to be reliable. And, a priori, it's likely to be highly highly unreliable.
It would be trivial to show, for example, an iterated chaotic system is sensitive to an x=pi initial state at "decimal places" that no possible physical computer could provide a discrete approximation of; and hence inferences made via this approximation would be, routinely, false. (This is, for example, why most "climate" models only predict a global mean temperature, and very little else).
So this all comes down to the need to formalise a non-discrete system in discrete terms, and worse in terms that are physically possible implement using electrical switches.
In this case, every output of the system including special designation of bit patterns is, a priori, profoundly suspect.
All of your objections continue to apply just as strongly to human mathematicians as they do to computers. But you apparently believe there is a difference between what the mathematicians can do and what the computers can do. This is false. Any problem that occurs in computers' representations of values will also occur in human representations of values.
Using your example, a system that is sensitive to differences so fine that they cannot be held in any realistic amount of memory is quite possible. But humans will have just as much trouble using it as computers do. If it is easy to show that x=pi in particular causes trouble, computers will find that easy too, using the same tools -- symbolic computation on pi -- that humans do.
The fact that computers have discrete internal representations is not relevant to anything. All human mathematics is also performed using exclusively discrete representations.
But I can at least now see why you're attached to extremely strange notions about, eg., floats being sufficient representation for mathematical reasoning. Ie., some article of faith that "computers" must be capable of everything.
There is no "symbolic computation on pi" that arent rules of inference created by people. We arent born with these rules, we create them. So if we havent yet created them, there's no sense in saying any actual computer is capable of anything. Actual computers are merely implementations of rules we'd have to create.
The process of conceptualising the world is, in my view, continuous and non-cognitive. One example of it is in the generative capacities of the imagination, which presents situations as wholes and it's latent space imv is continuous -- having to do with the structure of the sensory-motor system.
In any case, regardless of whether you believe animals have access to a continuous reality which cannot be formalised in discrete mathematics, we arent talking about whether there are possible computers which can reason this way -- we're talking about actual computers. (Though we have no reason to suppose there are such possible computers, and proofs against such things, ie., the non-computability of the reals).
It's relatively trivial to show that all existing computers are woefully incapable of a vast amount of things. Consider, only, the exponential space complexity of storing the parameters of a chaotic system. In any existing computer, we'd need an electronic system the size of a planet merely to track what's going on inside an atom.
It requires vast arrays of machines to track surface properties of particles interacting in the LHC, for example.
Yet, of course, we can formulate QFT. There are a near infinite number of such "existence proofs" of the power of animal mental capacities: AND NOT A SINGLE ONE! Of machine capacities.
No existing actual computer has ever created a system of concepts to formalise a hitherto unformalised domain. No one has even solved the problem of how it would be possible for a machine to do so (ie., the framing problem).
This makes actual computers, and all possible ones we can presently even imagine useless for open problems with unformalised domains.
The only role a computer can play here is providing an implementation of a discrete aproximation we have created, and this aproximation is woefully inadequate to the task. Even using a computer here is just a means of improving the power of human speculation.
In any case, this article of faith in the power of discrete mathematics and the electrical systems which we use to implement it, blinds you to the overwhelming and woeful inadequacy of all existing systems.
To the point you're even defending floating pt representations of infinity. If you really wish to cling to that religion, you're going to have to get better at choosing which hills to die on. Saying floats here are a sensible means of representing problems in continuous mathematics is absurd, and discredits your views greatly.
The only computers you should be defending here are "presumably possible" ones, yet to do be defined, yet even to be specified.
0 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ 7FF0 0000 0000 0000 (base 16) ≙ +∞ (positive infinity)
1 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ FFF0 0000 0000 0000 (base 16) ≙ −∞ (negative infinity)
This struck me at the time as a very powerful statement, yet unexpected, since very much not what most people expect from mathematics. After all, it's supposed to be a field where there is such a thing as a (most of the time) reachable truth!
If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks.
I hate this sociologist view that everything depends on the social consensus. Going extreme with it is how you end up in a 1984-esque society:
Anything could be true. The so-called laws of Nature were nonsense. The law of gravity was nonsense. ’If I wished,’ O’Brien had said, ’I could float off this floor like a soap bubble.’ Winston worked it out. ’If he thinks he floats off the floor, and if I simultaneously think I see him do it, then the thing happens.That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry.
Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented.
That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.
My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world.
Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection.
Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.
I suspect an engineering professor would be even more compelled to highlight how social consensus is the foundation of everything else that follows. You only have to read the surface layers of this or that performance debate to see how it is unfortunately the case. "Performance" is a fluid term that doesn't mean much of anything on its own -- and people arguing that they've juiced another drop of performance juice from this or that application's fruit are often just talking past each other in terms of priorities or perspective.
The symbolic manipulation at the heart of mathematics is a byproduct of language -- another system beholden to social consensus. It's inescapable.
There are also ways to philosophize about the nature of things in order to frame math as a human construct. I think both views can be simultaneously correct
For example, set theory. If set theory allowed self reference you could have set R, a set of sets that do not contain themselves. Would R contain R? It can’t, but it can’t not either. It’s self contradictory. The solution, reached through consensus, was to restrict the definition of the set to not allow sets to reference themselves [1].
As I recall, there was a paper by Robert Tarjan where he observed that most paradoxes are from self referencing. Soooo, rule out self referencing and will rule out most paradoxes!
But not everyone believes that 2+2=5, isn't it? I mean, if you criterion to separate truth from lies is the reality, then now you are talking about a counterfactual reality without proving that this counterfactual reality is possible. So this your statement is unproven, I'd say it cannot be proved.
I can argue, that if world believes that 2+2=5 then it makes 2+2=4 to be false. Just think about it. Try to imagine a plausible counterfactual reality where people believe that 2+2=5. They probably would believe that succession of numbers goes like this: 1, 2, 3, 5, 4, 6, 7, 8, 9, 10, ... And in such a counterfactual reality it would be very strange to believe, that 2+2=4. You could if you liked, and you could build a mathematics around it, but you would have a lot of problems of communication with others.
In math logic we can define systems of symbols and logical connectives and deductive rules. We can argue for the correctness of a logical system but we can not do it in the same system; we have to go "meta". Similarly a formal system could be a model of something "real", but the correspondence would be beyond mere logic.
A more accurate term might be stories rather than truth.
That depends on if one of the rocks breaks in half as you throw it onto the rock-pile or not. And also if the resulting piece knocked off is large enough to pass your fuzzy and contextual distinction between "rock" and "pebble".
But IMHO, arithmetic such as counting numbers and 2 + 2 = 4 are not part of the natural world. If I have a rock and another rock, I can with minor effort tell them apart: they have different weight, size, shape, density, composition etc. They are each unique individual assemblages of huge numbers of atoms in distinct never-to-be-repeated arrangements. In what way are these 2 unique things "the same" ?
If I have an apple and you give me a frog, I have an apple and a frog. They're not the same. If I have apple A and you give me apple B, do I have 2 apples? I have unique apple A and unique apple B. We can pretend that they're the same if you like, but that category is in our thinking, not in the world, and we also know that we can also notice differences between them.
tl;dr the natural world is not fungible, but behaving as if it is, is a convenient abstraction for mathematics and commerce, not a property of the natural objects.
If it wasn't a property of natural objects (in some way), then how could our predictions work so well in the real world?
It works for electrons. But when was the last time that you interacted knowingly with a single electron?
For example, all the predictions that make us capable of building skyscrapers that don't fall down for centuries. Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them? In terms of their behavior under particular circumstances, they are the same.
Bricks are in the category of "made objects" not natural objects, which generally means that they are _designed_ to come off a production line as similar as humanly possible to the other products. "My iPhone is physically interchangeable to yours" is a statement about the huge efforts of industrial manufacturing to standardise matter, not about the natural world.
Bricks too have quality thresholds that they have to meet or exceed. That alone should tell you that treating them like integers is a convenient abstraction, nothing more. The sibling comment has it right: counting bricks is a great model, but confusing your model for reality is still an old error.
No, because of a dense, interconnected web of other social truths (the rest of the arithmetic model), the relative error of this one truth/model is negligible.
However, confusing your model for reality is a fallacy perhaps older than time.
> However, confusing your model for reality is a fallacy perhaps older than time.
I think that the human perception the world is a robust enough model to be equated with reality without issues. If you go down the path of denying perception, you might as well go full solipsism, in which case it doesn't even make sense to discuss reality at all.
The model is "good enough" for the purpose of creating a building, but that doesn't make the act of "creating a building" any less real, nor the underlying rules that govern matter any less true. Our descriptions are not real - the rules themselves (which we may not know exactly) are real.
Therefore, mathematics - the set of rules that corresponds to how reality works - itself exists in reality regardless of social consensus. Society can't change them by making a different consensus.
People have agreed on how the plus sign should be defined when it comes to adding two integers, and they've agreed on how to write integers, in which case 2+2 is always four. The only way to change that is to change the meaning of at least one of the four symbols "2", "4", "=", or "+" to some private meaning which no one else shares -- e.g. to distort the language in order to sow confusion, which the use of agreed upon meanings for symbols is designed to eliminate.
Moreover this idea that a sentence changes its truth value the moment someone changes the meaning of one of the words in the sentence is not some deep statement or interesting observation, it's just pointing out that language is useful only if people agree on the meaning of the words.
You’re asking a simple grammar question under the impression it’s an unsolved science question.
Everyone knows what a rock is. Everyone knows what I mean by "throwing two rocks on a pile of two rocks and getting a pile of four rocks" - almost everyone has done it countless times as a child. The "What's a rock" comment is just pointless philosophizing, masquerading as profoundness.
Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. It also is much less cumbersome than it used to be, and will continue to improve quickly.
I can sit down and prove something using a tool like Isabelle, and I will be as sure of its "truth" as I can possibly be, and it really doesn't matter what other people, mathematicians or not, think about it. That's the beauty of it.
Of course, you could say my belief in Isabelle is also a social construct. Except it is not, I know exactly how Isabelle works. There could be issues with Isabelle, but these issues adding up to make my proof wrong are very unlikely, especially in addition to my independent understanding of the proof.
But of course, it is much nicer if others can see the same truth that I do, and for this, computer-assisted proof is actually great, because it allows to understand and trust in the high-level structure of a proof without having to verify every little gritty low-level detail.
I think you are mistaken. The idea that math proofs are a social construct relates to, in my view, much deeper ideas than you seem to think [1].
It is not just that convincing other mathematicians that a proof is correct is a social process, but also that the reasoning on which any proof relies, even if it seems unassailable, even if built into an automated checker, is still a product of the human mind. Usually there is a level of logic that can challenge even what seems so basic as to be fool-proof.
Take the proof that the square root of 2 is irrational. The proof relies on a contradiction that arises if one assumes the root is rational, but one can imagine a logic system where such a contradiction does not imply that the original assumption is false. How possible, you say? It's all math, where one is allowed any starting assumptions, and works out the implications of those.
But, there is something deeply satisfying about thinking that contradictions are (or should be) impossible in our universe, and so this "proof" seems solid.
Personally, I don't think intuitionism makes much sense on a fundamental level beyond being an elegant mechanism in certain situations. If you tell me that it is false that A is false, then certainly A is true. Anything else is really mystic mamboojamboo and not clear thinking. But that's just my opinion, and I am not gonna force it onto you, because this is not something I can prove in a proof-assistant, but just an opinion, and as such a social construct.
I am not really dogmatic about this. You might be able to use intuitionism for things that cannot be done via classical reasoning, for example extract programs from a proof. I have yet to see an example where it is not simpler and more straightforward to just prove an executable program to be equivalent to the specification using classical reasoning.
My point of view is the following: If you are not able to (eventually) make your case within a proof assistant, then what you are trying to tell me is not math.
The truth or falsity of that statement in a model is independent of who observes that model.
The only “social construct” you’ve described is which model to use by default — literally, a notation convention, nothing semantic.
> But, there is something deeply satisfying about thinking that contradictions are (or should be) impossible in our universe, and so this "proof" seems solid.
Our universe either does or doesn’t, depending on which model best represents it — we’re only debating what assumption to make about an unknown.
I think you are mistaken.
I used to be very troubled by the notion that no single set of axioms really can be agreed on to do mathematics, but I have been convinced finally that the truth of the matter is a very subtle point; that the truth of mathematics is absolute; it merely is not finitely-axiomatisable.
With a sufficiently weak proof system, clearly we can conceive of a system where the irrationality of the square root of 2 is not provably true, but no consistent proof system can prove that the square root of 2 is rational. Certain mathematical constructs, indeed most(*) of known mathematics, that which is constructible by constructivist methods, are irrefutably there in any consistent mathematical universe, and in that sense, true.
Yet, no single axiom system can encompass all mathematical truth, as is well known from Goedel's theorems, but neither does that mean that the set of axioms to be worked with can be arbitrarily chosen. The chosen set of axioms must be consistent. The question, then, is whether consistency of axioms can possibly be an objective fact; and even though for any sufficiently strong set of axioms, its consistency cannot be proven in of itself, it consistency can in fact be objectively established - objectively established, but not finitely established.
My evidence for this perspective is Scott Aaronson's construction of a Turing-Machine encoding of the ZFC axioms. What the construction of this encoding implies, is that the revelation of the uncomputable busy-beaver (BB) function for value 8000, which is a finite, well defined, and an objectively irrefutable, albeit unthinkably massive, number, constitutes a proof of the consistency of ZFC axioms. A similar procedure I believe can be applied to any set of axioms that one wishes to work with.
The part where the magic occurs, I believe, is in the uncomputable nature of the BB function, by which it is possible to finally and objectively establish consistency of sets of axioms. Uncomputability amounts to the acknowledgement that though something may always be well-defined, there is no finite method to encompass its values; that is the nature I now take of mathematics as well.
So the very foundations of mathematics, the logic we use behind our proofs, is inherently a social construct that arose out of Greek philosophy as it was adapted in the west.
I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical structure itself. The process is social but the validity and mathematical structure itself is not.
Computer science has led to an explosion in different logics. We've never had more logics/formal systems than we do today, but whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus.
The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which nuances of meaning between the Platonic Ideal and our concrete reality are actually reflections of the truth or corruption of it.
Many do not seems to see the fundamental issue at play here, and another way to think of them is what you have hinted at: the role of language. There is no objective way to nail down the meaning of words, like "consistent", "proof", "equal", etc.
Suppose one wanted a maximally rigorous definition of "equal". Does it mean two things that cause people to think of the same thing when they are mentioned? Does it mean two things that occupy the same position in space at all times? It is actually a difficult concept to define rigorously.
This is not to deny that there is an objective reality. But that reality is highly contextual and multi-faceted. We cannot be 100% exact in defining that reality using language (even a math language), and this is where the social nature of that reality becomes apparent.
The role of proofs are in creating, as far as possible, as rigorous a shared context for the reality being described.
1) x = x
2) x = y => P[x] => P[y]
In my opinion, there is a mathematical reality, which is shared by everyone, even by those who don't believe in it :-) For example, a logical system exists in that reality, and you can either derive a theorem in that system or not in this reality. I don't think it is possible that there is a third possibility. I don't think it is possible that I have a different reality from you in that respect. This reality is not socially constructed, it just is. Intuitionism will tell you that because you don't know if a certain theorem is derivable, it is in some sort of hybrid state until we know for sure via a intuitionistic proof or a counter example. I think that is bullocks. Either there is a proof or not. Either there is a counter example, or not.
Beyond that, extending this mathematical reality, there is a wider, not as easily accessible reality. We can try to understand that reality by modelling it via certain assumptions, and then applying our mathematical reality to those assumptions. I believe the mathematical conclusions we draw from this will be real to the extent that the assumptions are true; but of course you cannot ever be sure about those assumptions, and so you cannot be sure about the conclusions. But if you notice that your conclusions do not hold, you need to challenge your assumptions, not your mathematics.
In short what it usually means (the exact meaning depends on the model under consideration) is that there is a binary operation "=", such that "x = x" is a theorem, that is evaluating "x = x" will evaluate to "true" for any object x in the mathematical universe. Furthermore, for any unary proper operator "P", and any two objects x and y of the mathematical universe, the expression "(x = y) => (P[x] => P[y])" will also evaluate to "true". Here "=>" is another binary operation called implication, which has some special properties outlined in the link. P[x] denotes the application of the operator P to the object x.
Edit: Oh, forgot to add the third axiom for equality (it is actually more an axiom about "true", but uses equality):
3) A => (A = true)
What this means is that for any object A of the mathematical universe, if you evaluate "A => (A = true)", you obtain the value "true".
I think you're conflating universality and objectivity. Those terms you list all have objective definitions, but the specific characteristics they have in any given logic may differ. That means they are not universal, but that doesn't make them non-objective. Objective typically means "mind independent".
Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change. That is an informal but objective understanding of equality.
What you're recognizing is that equality can have different logical properties in different contexts, where "context" can be understood as the formal language we're using, ie. it's not universal. But it's role in any given logic is always the same and not dependent on the provers mind state or his surrounding culture, ie. it is objective.
Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program).
So what logics a culture or species may use or find interesting, and the process by which they explore these systems are socially contextual, but the structures themselves and their internal consistency is not socially constructed. A culture can certainly believe a formal system they use to be logically consistent, but that's no more interesting a statement than that some cultures believed that Thor caused lightning. In other words, they could just be wrong about the consistency of their arguments.
> This is not to deny that there is an objective reality...
So I do agree that object reality exists, contrary to what you imply in your response.
However, there is no word in human language that has an objective definition. If you think this is so, maybe you have not thought or read enough about the subject.
On the subject of equality, you said: > Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change.
You're making my point. Consider the vagueness of what you have written: "...some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change." That doesn't sound very rigorous to me, and in fact I could argue this is not what I understand by "equality" at all.
Now you could make the point the definition can be made rigorous for a very specific context, but this is still not 100% possible. To show this, you can attempt to give me a specific definition of equality of a specific context, and I will show that it still admits of multiple possible meanings.
Could you elaborate on that? Any references?
This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc).
Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...
Could you elaborate on what such a formalism where infinite structures are not expressible might look like? It sounds an intriguing notion, though I'm unable to quite get at it.
A Defense of Strict Finitism, http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf
This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.
Even in our commonly used logic system, what does false actually mean?
false = (∀x. x)
Nobody is questioning that there are different possibilities for logical systems, and different possible semantics for them. The choice of logical system is up to you, and ultimately, part of your assumptions. A proof assistant uses a fixed logical system, and your mathematics needs to fit into this system to be expressible in the proof assistant. Apart from my favourite logic, there are several well-known logical systems commonly assumed to be able to represent all of mathematics more or less faithfully: first-order logic, simply-typed higher-order logic, and dependent type theory.
What I like about my favourite logic is that I believe that all other logics, even quantum logics, can be expressed in it in a straightforward way. This has the advantage that once you expressed your logic in this way, you get the meaning of it for free. Of course, you still need to make sure that this is the intended meaning.
That is kind of obvious, isn't it? But some people need more convincing than others :-)
But I guess the point is that almost no-one does this. I would guess that if someone tried to formally verify every published paper out there (or even every textbook), they would uncover a large number of gaps. Very few of those gaps would be unfixable, and very few results would turn out to be incorrect, but the possibility exists.
Everyone thinks proofs are this holy grail and totally rigorous, and they are on a certain level. But the idea is floating around that Mathematicians are infallible when in fact lots of proofs in highly complex areas of mathematics are NOT 100% perfectly rigorous. They contain a lot of skipping, because "it's trivial" and consensus.
This approach may work very often, but there is a danger that sometimes it doesn't work and things get overlooked. Since mathematics is done in a bottom-up approach, at some point some fundament may or may not turn out to be wrong, which endangers parts built on top of it.
The whole movement of rigorous automated proof systems is to prove mathematics from the very bottom to the very top in a 100% rigorous and verifiable way.
Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation.
That's a very subjective POV, and perhaps one that varies by area of math. Many computer proof developments are more cleanly refactored/abstracted than the manual equivalent, because it's so easy to refactor a computer proof without worrying that the new proof might fail to prove the same statement.
Couldn't we also interpret this fact that computerized proofs are currently often very unelegant as strong evidence that not a lot is understood about this topic and thus doing such "ugly" computerized proofs is the best we can (in most cases) currently do?
Science at the boundary of human knowledge is often quite ugly; as our understanding of it grows, it often becomes more beautiful and elegant.
If one can afford it, they can travel to Paris and check themselves. The same with mathematical truth: if one has means (time, intelligence, access to training), they can check the proof themselves. Otherwise they need to trust the consensus.
So again, is the truth in mathematics just a social construct? In some sense, I guess, but probably not the one some people might assume hearing such a statement.
How about checking with a GPS?
A social construct has nothing to do with simple facts about the universe. And whether the Eiffel tower exists as an object at a particular spot as indicated on maps is such a fact. And if there were maps that would place it elsewhere, those maps would be a lie. Even if the every single map ever made and every other person would deny that there is such a tower at that position one could still go there and check for oneself.
Maybe you are talking about the name? The fact that we call it the Eiffel tower? Well, that tower has a history and again one could lie about the history, who built it, how it was historically called as a matter of fact etc. But an observer would have seen who actually built this tower. It's a fact.
Fred Moxley has (what seems to me like, but what do I know?) a nice proof of the Riemann conjecture that he got by quantizing the problem. But nobody will read it, because mathematicians don't like that method. It might be right or not, but it anyway doesn't tell you anything surprising about prime numbers, so nobody can be bothered.
The truth is.
The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is.
There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true?
Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans.
So for example, if the proof doesn't convince other mathematicians, then it's not a a proof.
Or it might convince other mathematicians and later turn out to be wrong after all.
For more on the practical aspects of math, I highly recommend The Mathematical Experience.
https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav...
I read it in German:
https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...
> The truth is.
Yeah, that is how it feels like nowadays, however the truth is bound in a narrow set of assumptions. These assumptions are bound in reality even in mathematics (One apple is one apple, you add another one, you have two). And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct.
The details of how a truth is communicated is in a sense a social contruct, because communication as a whole is, however nobody would call it like that. It is maybe a small reminder that meddling with language for no apparent reason is a warning sign, but this is going a bit off-topic.
To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.)
There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.
I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem.
You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.
So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.
This is utter nonsense. An apple IS an apple. If I put one on a table, then obliterate every human being that’s capable of sensing it, the apple is utterly unaffected.
If aliens come down and experience the apple differently than we would have, that doesn’t change the apple one bit.
The world is a sea of particles and energy which behave according to certain patterns (both fundamental laws and emergent behavior). Some of these patterns are pertinent to us, so we name them, giving rise to a category. "Apple" is such a category.
The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.
If aliens then visit who can't eat the apple and aren't interested in botany, they may simply choose not to distinguish between apples and pears, or apples and any other fruit, or even apples and any other form of organic material. The same clump of molecules is there, but the categories it belongs to have changed.
Correct. If all humans cease to exist, the human-made abstraction of "the apple" passes away. But an apple--the one on the table--is completely unaffected.
The difference here is that between an abstraction and a concrete instantiation of said abstraction. The abstraction of "the apple" is indeed a subjective construct which would pass away with humanity. But OP said "an apple". "An apple" is a part of the physical universe. It is, like you said, a clump of molecules. Molecules that can be measured objectively.
An apple is no more subjective than the orbital period of the earth or the length of the standard meter.
Sorry, but no. Any species capable of actually creating some kind of math will have some mathematical structure isomorphic to the integers. If cats can't count, then that just says that cats are not capable of creating some kind of math.
Secondly, discretization absolutely is universal. It's literally in the laws of physics for one (particles are discrete, energy levels are discrete, etc.). For another, are you suggesting a physical alien species will have a continuous number of appendages, or organs, or that their population will somehow be continuous? I frankly don't see how you can possibly escape aliens capable of math developing a notion of basic counting.
Secondly, what we know must be bound by what we've observed. You can imagine any sort of being you like, but that doesn't make your imagined creature logically coherent or physically realizable.
Any physically realizable intelligence must:
a) Be differentiable from its environment: that means it must have some enclosing boundary separating an inside that's different than an outside.
b) Have internal structure: intelligence by necessity is structured thought. Structured thought entails differentiable physical structure to hold structured thoughts. Such structure by itself is necessarily countable, being made of matter.
It was clearly not about the apple, but about distinct entities with similar features. Now for another being these might not be similar, but something else probably is, disregard of different dimensions, different senses or the likes. As the observed reality for an alien species or actually any other species on this earth is different of course.
> You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.
> And you can't prove a subjective experience objectively.
I think the misconception comes from the fact that you need basic assumptions to build an abstraction. One of the most basic assumptions is that something like a shared reality exists and we're not for example in a virtual world or a dream.
You can happily deny this shared reality, however I would not necessarily encourage you to touch fire (literally and figuratively speaking).
> Yeah, that is how it feels like nowadays,
It's always been that way. (Again, I really recommend the book[1] ). And it's hard to see how it could be otherwise.
(Also depends a little about what exact mathematical truth we are talking about and whether you are a Platonist or Constructionist)
That doesn't imply what either the recent proponents or the critics seem to think. It does not at all imply arbitrariness or that anything goes.
> however the truth is bound in a narrow set of assumptions.
Yes, it is. Again, something being a social construct does not make it a free-for-all. More the opposite, because the constraints are socially enforced.
> And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct.
Mathematics ≠ Reality. Science is about reality, but scientific truth is also a social construct (see Popper), and highly constrained by reality (ibid).
[1] https://www.amazon.de/Mathematical-Experience-Phillip-J-Davi...
Mathematics is about describing possible worlds. Given these assumptions (including the rules of the game), what follows?
Science is about figuring out the real world. The real world has no obligation to be describable by mathematics. That it is so describable is fortuitous.
"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."
https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
I didn't mean "necessary" in the sense of modal logic, but in a more colloquial sense. Physical laws as known to us today are described using mathematics. We know of no other viable ways to describe them. In that sense, (a subset of) mathematics is necessary for this purpose.
It was just meant as a restatement of the concept of unreasonable effectiveness of mathematics in a more natural language.
> The real world has no obligation to be describable by mathematics.
There is no obligation, but that it is so in our particular world is undeniable fact.
Conjecture. A mathematical universe is consistent with everything we know, in which case math is literally the study of reality.
1. Math can describe lots of universes that differ from observation. For starters, just look at all the different kinds of space-time there could be.
2. On top of that, even the math that we have that describes our current universe is...tricky: the math at the core of our two best physical theories (General Relativity and Quantum Mechanics) is inconsistent, i.e. they both cannot be true at the same time.
Lots of our greatest minds have tried to find a mathematical formulation that makes the two compatible, but so far none have succeeded. So there may not actually be a mathematical formulation.
3. And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance, and one we might be finding the limits of (see (2)).
"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."
https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
This has all been hashed out already. Look up the mathematical universe hypothesis and Wolfram's Ruliad for two different but similar approaches. The basic idea is that all consistent mathematical objects exist, and we simply inhabit one of them.
The fact that QM and GR are inconsistent with each other is not an issue because they describe different universes in mathematical space that are close but not equal to our own.
> And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance
That too is a conjecture. There is exactly zero evidence that nature is non-mathematical, but people who keep saying this are basically insisting that it's effectiveness at describing reality is simply a continuous chain of grand coincidences. To say this is statistically implausible bordering on impossible is the kindest way to put it.
Would that invalidate (or not), the statement that 1 + 1 = 2?
But when we say "proof", we usually don't mean "fully formal proof", and there are good reasons why:
1. Fully formal proofs didn't exist for most of the history of mathematics. 2. School by and large don't teach formal proofs, and most students are probably not aware of the existence of formal proofs. 3. Even today, most professional mathematicians never write formal proofs, except perhaps as an exercise during their education.
So what do we mean by "proof" if it is not an argument that is exhaustively traced back to axioms? It's really no more than "that which the teacher/other mathematicians will accept as a convincing argument". When you learned a proof of the Theorem of Pythagoras in high school, you almost certainly didn't learn a fully formal proof. You learned a proof that, at some point, just implicitly got "cut off": the proof tree didn't work all the way back to the axioms but stopped at some point where the argument would become too tedious to continue laying out in full (without even being told that the proof got cut off: your teachers perhaps even told you that this was a rigorous argument).
To write such a proof, you need to judge where the acceptable cut-off point is, which is entirely based on what other people will accept as good work. Hence, a social construct.
edit: if you're not convinced that the proofs you learn in school/university aren't fully rigorous, I warmly recommend trying out a proof assistant like Coq, Agda, or Lean. Try to encode some well-known piece of mathematics. Euclid's Elements is a good candidate: working through it fully formally, you'll find huge omissions in the Elements immediately.
I don't think you can make this claim really, precisely because the logic we accept as, well, logical, is a social construct. Different cultures across different places have had different ways of accepting what is valid in an argument. The methods of logic we consider valid are themselves social constructs, basically.
When you learn math at school, at first you're just told that these things are true, and so it becomes an accepted truth how addition works.
Only much later can you go and verify the proof from the axioms.
Similarly, if someone today relies on Fermat's last theorem to hold for some of their own work, they're unlikely to have verified the entirety of Wile's proof. Rather they lean upon the experts who have, and thus have accepted the truth that the proof holds.
(Yes, proofs relying on AC are arguably true even if you don't accept AC, but as a social reality some sets of axioms are considered valid bases for work and some aren't, you can keep adding stronger axioms to ZFC to prove more things more easily, but how far you go with that before it stops being interesting is a matter of opinion)
"the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2."
Most of us conflate arithmetic with mathematics. In arithmetic, things start getting 'conceptual' as soon as we no longer can map certain measures and operations to a realizable physical construct. At that precise juncture, math becomes a semantic system and is therefore subject to social consensus.
For example, consider introducing infinity, or even zero, into 'shopkeepers' sense of numbers. Before, you could never add something to a number and end up with the same number, but now 0 + 0 = 0, and a + \infty = \infty . And to the shopkeepers' surprise, some mathematicians may even argue over it.
An interesting meditation here on mathematics itself, which is also simply certain rules, and not others.
Merely invoking social construction ignores this difference, which is the essential difference, between mathematics and, say, hide and go seek.
So how do calculators and computers work then?
It’s simple, repeatable and therefore programable - but still a social construct.
And the rules they created to make that definition only apply in the Solar System, not to exoplanets around other stars nor to rogue/free-floating planets.
And Ceres, Pallas, Juno, and Vesta were considered planets for over half a century.
What about "US citizen"?
I don't think so.
Which means we can't say the existence of a word means its the result of a physical assumption.
When you wrote "planets, stars", did you not assume those two are different? But originally, "planet" meant "wandering star" - a planet was assumed to be a type of star.
Would that invalidate (or not), the statement that 1 + 1 = 2?
And I hate to break it to you but all “axioms” are social constructs that we use to create hopefully useful models.
> but all “axioms” are social constructs
Again, they are mathematical constructs. Axioms don't have anything to do with society. I can construct any axiom I want, and derive true statements from it. Perhaps the only "social" thing here is what axioms we generally deem interesting enough to investigate - this does not change the facts derived from them.
> that we use to create hopefully useful models.
I think you underestimate pure mathematics :)
> definition of addition
What do you think any definition is other than the creation of a socially shared construct?
> I can construct any axiom I want, and derive true statements from it.
Let’s test this. Axiomatically I am always right! As such, I derive that it is illogical, dare I say irrational, of you to disagree with me. A truer statement has never even been written! Facts!
Hmm… turns out you disagree that it’s a true statement. Why? I had an axiom! You don’t agree with my axiom? If that’s allowed, I guess I’ll have to convince you (and society) of my axiom!
A definition is simply a definition. It doesn't have to be social or shared. I can define addition in any manner, and in fact there are many other definitions of addition.
> Let’s test this. Axiomatically I am always right! As such, I derive that it is illogical, dare I say irrational, of you to disagree with me. A truer statement has never even been written! Facts!
You have a very interesting misunderstanding here. A fact is true only relative to an axiom. So yes, relative to such an axiom you are right, similarly how relative to the axioms of the field of real numbers the statement 1 + 1 = 2 is true.
You can construct any axiom and derive facts from it. I can construct axioms such that 1 + 1 = 0. That doesn't make the statement 1 + 1 = 2 in the real numbers any less false!
I think the issue is if you call every type of thought and communication "social construction" then you don't end up with anything useful.
That's a different thing.
Your teacher was just a sophist.
At some point the thought occurs to me. Which is more likely, there's a guy who knows all the same stuff I know, is in the same awkward work situation I am in, happens to find the exact things I am looking for, OR I am having a psychotic break, this guy is my delusion, and all those things he does is actually just me doing it? After thinking this I started to realize, I had never really seen this guy outside the apartment. No one else I knew from before had ever seen or knew of this guys existence.
One day I'm idly humming a tune that got stuck in my head. You might recognize it as "Battle hymn of the Republic." But, there's actually 4 prominent songs in American history with this exact same tune. The others are "John Browns body", "Blood on the Risers" and "Solidarity Forever". My new roommate walks in and starts singing the words. But how did he know which one I was humming? It wasn't the obvious well known one! No, surely I am going mad.
My roommate had mentioned that he lived in Russia until he was 8 and could speak basic Russian. I do not know Russian. I ask him to teach me about Russian grammar. He agrees but then changes the topic. I push the issue again latter that day. He once again agrees to teach me some Russian and then proceeds to divert attention elsewhere again. I ask him to teach me some Russian. He pushes it off yet again. Whereas before the thought was idle, the evidence keeps on growing. I'm having a psychotic break. This guy can't be real.
We are sitting around one day. My roommate points out that all of us sitting in the room have hazel eyes, and that this is the rarest of the eye colors. They then proceed to pull up the statistics and crudely calculate the probability of this happening (pretending our genetic demographic is unrelated to the circumstances that led us all to this room). The result was some outrageously small number, less than a tenth of a percent. At this point I'm pretty sure my own delusion is taking the piss out of me, actively shoving the implausibility of his own existence in my face as a joke.
It turns out all of this really was amazingly coincidental. As weeks went by, guests at the airbnb would come in go, we would meet each others friends, and eventually there were enough people who also acknowledged his existence that I am now convinced he is real.
So I pose it to you. Was my roommate real, was everyone involved in this story a figment of a madman's imagination, or am I completely making up this roommate story to make a point? The answer is, reality is shared consensus. If you all are also convinced that this person existed and these events transpired, then we share a common set of facts. If there is no shared consensus, then he only exists for me. Perhaps there is some underlying truth beyond the shared consensus, but shared consensus is the instrument we use to measure realness. At some point, there is no difference between "every multimeter says this battery is 9 volts" and the battery actually being 9 volts.
I'm going to relay another story. Neils Bohr used to keep a horse shoe nailed to his door. When asked, he would say its for good luck. One day someone asked "do you really believe that?" He responds "No, but they say it works even if you don't believe in it."
Why believe quantum mechanics over lucky horse shoes? If everyone chooses lucky horse shoe theory, does that become reality? If powerful interests in government start forcing everyone to adapt horse shoe theory, does that make it real?
Thus I arrive at a truly bothersome set of contradictions. Reality is shared consensus, but reality is also the set of things not subject to popularity. There is no truth only power, but also the essence of science and math is that truth does derive from authority.
One day I will reconcile these. One day.
This is garbage.
Everything under this definition is a social construct and as such why would you only relate it to mathematics?
The rock I'm holding is a social construct. Deep.
If they want to get stoned and talk about the meaning of life cool, but it's beneath a math professor (Who's not at home getting stoned)
Following it logically you quickly find murder, rape, genocide being bad are just social constructs. And why exactly should we follow social constructs? Lets all go and start the next FTX because everything is just a social construct so who cares?
And we've just rediscovered nihilism like the other 120 billion teens did.
Going from self-reflection to nihilism is a pretty big overreaction.