[HN Gopher] Computer proof 'blows up' centuries-old fluid equations
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Computer proof 'blows up' centuries-old fluid equations
Author : nsoonhui
Score : 166 points
Date : 2022-11-18 11:19 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| paulpauper wrote:
| How is it even possible to learn this much math or organize such
| complex ideas. Undergrad comes nowhere even close to this. I
| guess the leap occurs in grad school. But even though, this is
| way more advanced than 99.75% of math papers I have read.
| kzz102 wrote:
| Computer assisted proof is not a new idea. The most famous
| example is the four color theorem, which (as far as I know) does
| not have a humanly understandable proof. For this particular
| proof, it looks like the computer assisted part would be similar
| to writing down hundreds of pages of checkable inequalities. One
| way to do this is to use interval arithmetic, which always
| guarantees the answer is within the a certain given interval.
|
| 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.
| wwalker3 wrote:
| The question that the referenced paper (1) is trying to answer is
| "do the 3D incompressible Euler equations develop a finite time
| singularity from smooth initial data of finite energy?" This is
| an important question in the theory of nonlinear partial
| differential equations, but is probably not as relevant to real
| fluid flow as a lay reader might imagine.
|
| 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.
| pencilguin wrote:
| I guess this answers why you can't just try it with liquid
| helium: even _that_ isn 't ideal enough.
| nuclearnice1 wrote:
| What a wonderfully informative and educational comment. Thank
| you.
|
| 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.
| wwalker3 wrote:
| The incompressible Euler equations model a fluid as a two-
| valued field. This means that at every point in space, the
| field has two values, density and velocity (1).
|
| 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.
| vba616 wrote:
| >The incompressible Euler equations model a fluid as a two-
| valued field. This means that at every point in space, the
| field has two values, density and velocity
|
| 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?
| wwalker3 wrote:
| The density can be constant, but it doesn't have to be.
| If the density field starts out with some variation in
| it, then those variations move around as the fluid flows.
| Incompressibility just means that those density
| variations can't get bigger or smaller, they can only
| move, shear, and rotate.
| NotYourLawyer wrote:
| It's when some physical quantity of the simulation becomes
| infinite. Pressure, particle velocity, etc.
| TEP_Kim_Il_Sung wrote:
| That usually indicates a phase change, with a separate set
| of applicable equations.
| isoprophlex wrote:
| I think (not a physicist), simply put, an infinity or NaN
| value. As these are step-wise methods, having such values
| show up anywhere will seriously mess up subsequent
| calculation steps.
| prolyxis wrote:
| A simple example of a function with a singularity is
| f(t)=1/t. Note that at t=0, f(t) is undefined due to division
| by zero. On either side of zero, the absolute value of f(t)
| approaches infinity.
|
| 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.
| nico wrote:
| Thank you for the great explanation.
|
| 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)?
| davnn wrote:
| > This is an important question in the theory of nonlinear
| partial differential equations, but is probably not as relevant
| to real fluid flow as a lay reader might imagine.
|
| What kinds of problems does it solve to know an answer to this
| question? Honestly curious, please do not take this as
| offensive/dismissive.
| wwalker3 wrote:
| If mathematicians could solve these kinds of problems, they
| could answer valuable questions like "Will this equation
| always have a physically meaningful solution?" If the answer
| was "No", then we would know that the equation can't be a
| faithful model of reality.
|
| 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.
| civilized wrote:
| This raises a question I hadn't thought of before. Real-world
| fluid flow is ultimately well-modeled by the equations of many-
| body Newtonian mechanics, right (atoms bumping around)? Are
| those equations vulnerable to blow-ups?
| wwalker3 wrote:
| Pretty much any mathematical model of a real phenomenon can
| have some sort of singularity or discontinuity in it.
|
| 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.
|
| 2) https://en.wikipedia.org/wiki/Linear_elasticity
| civilized wrote:
| I'm asking about the actual properties of the equations,
| not if it's hard to do simulations.
| wwalker3 wrote:
| It's the equations themselves that are singular. When we
| write simulators, we usually have to paper over the
| singularities that are inherent in the math.
|
| 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.
|
| 1) https://en.wikipedia.org/wiki/Coulomb%27s_law
|
| 2) https://en.wikipedia.org/wiki/Interatomic_potential
| civilized wrote:
| The presence of a singularity in the force doesn't mean
| it will cause a blow up in the solution. Two positively
| charged point particles interacting electrostatically can
| be shot at each other at any angle or speed and blowup
| will never occur.
| vba616 wrote:
| This makes me think of:
|
| 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.
| throwaway81523 wrote:
| There are all kinds of blow-ups in Newtonian mechanics and in
| other equations of physics. The singularity at the center of
| a black hole in general relativity is a famous example. The
| ultraviolet catastrophe in classical thermodynamics was
| another. The presumption is that blow-ups in an equation
| indicate a mismatch between the equation and the true
| physical world, telling physicists to look for better
| theories, whose equations don't blow up. For the ultraviolet
| catastrophe, the mystery was solved through the discovery of
| quantum mechanics. For GR, it is still unsolved, and the
| solution is expected to come from a theory of quantum gravity
| that hasn't yet been invented, but is the target of tons of
| research.
|
| Here's a cool expository article about blow-ups in classical
| mechanics and elsewhere: https://arxiv.org/abs/1609.01421
| WalterBright wrote:
| Best comment I ever read on HN.
| wwalker3 wrote:
| Wow, I'm honored :) These days, I try to only comment when an
| article is really in my wheelhouse, but that's not very
| often, given my narrow interests in fluid dynamics and
| computational physics.
| WalterBright wrote:
| > I try to only comment when an article is really in my
| wheelhouse
|
| 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.
| graycat wrote:
| Yes, in pure/applied math, we know a lot about various cases of
| _approximation_. But in practice there are more cases of
| approximation, and, right, the Euler equations are another such
| case. Or, to be a little flippant, generally in applications to
| real problems, we look at a lot of the _features_ and throw out
| some, modify some, and actually honor some!!
|
| 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??
| Agingcoder wrote:
| I remember my math professor at university telling me that truth
| in mathematics was a social construct, and that nothing was true
| until a social consensus had been reached between mathematicians.
|
| 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!
| faraaz98 wrote:
| Is it because maths is "incomplete" ala godel incompleteness
| theorem?
| samatman wrote:
| Mathematical truth is socially constructed, but using rules,
| and it is the rules, rather than the process of social
| construction, which give this process its power.
|
| 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.
| bheadmaster wrote:
| The fact that mathematics gives us power to predict events in
| the real world makes it independent of social consensus.
|
| 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.
| SideburnsOfDoom wrote:
| > 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.
|
| 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.
| bheadmaster wrote:
| > 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?
| SideburnsOfDoom wrote:
| What prediction is that? Are you arguing that two rocks
| or apples or people are actually "can't tell them part"
| identical?
|
| It works for electrons. But when was the last time that
| you interacted knowingly with a single electron?
| bheadmaster wrote:
| > What prediction is that?
|
| 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.
| SideburnsOfDoom wrote:
| " In terms of their behavior under particular
| circumstances" is very specific. "The maths is useful
| under particular circumstances" is not the same thing as
| "numbers are real"
|
| 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.
| musingsole wrote:
| > Does it matter that two bricks are not "the same piece
| of matter" if our predictions work the same for both of
| them
|
| 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.
| bheadmaster wrote:
| I don't understand how is "the arithmetic model" a social
| truth, when it clearly corresponds to physical phenomena.
| You can make a skyscraper that doesn't fall, and it
| exists regardless of whether other people see it or not.
| You see it - it's there. What is "social" about that?
|
| > 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.
| broast wrote:
| As far as rejecting perceptions i think going straight to
| solipsism is a big jump. We may live in a reality that we
| have no access to. Donald Hoffman's theories in this area
| are fun.
| guipsp wrote:
| We have a model of physics which is pretty accurate. The
| engineers who designed the skyscraper did not even use
| this model, they used a much simpler one, with known
| errors. Why? It is simply good enough(tm). But you can't
| claim it is even "true" when we know more accurate
| methods.
| bheadmaster wrote:
| My claim is not that the _model_ itself is true - I 'm
| claiming that the underlying mechanisms that rule the
| world are true and are not subject to change by social
| consensus.
|
| 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.
| ethanwillis wrote:
| What's a rock?
| bheadmaster wrote:
| What's a "what's"?
| ethanwillis wrote:
| Well for either of us to know, we'd need consensus on
| language, meaning, words, etc.
| edgyquant wrote:
| Nope. The idea of describing an object is built in to
| humans and likely all mammals.
| edgyquant wrote:
| An inquiry for a description of an object. Nothing in
| language is difficult it's finding the smallest
| abstraction to generate all the rules for all language
| that is.
|
| You're asking a simple grammar question under the
| impression it's an unsolved science question.
| jfengel wrote:
| That's actually an example of what OP was talking about. You
| have defined + as the operator that mimics what piles of
| rocks do, and defined numbers as counting rocks.
|
| 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.
| bheadmaster wrote:
| > 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.
| [deleted]
| jxdxbx wrote:
| You can construct a formal system with any axioms you
| choose. It is not arbitrary that some of these systems
| turn out to be useful in modeling the world. But there
| are other systems that are only dry exercises in symbol
| manipulation that may be of little use to physics or
| engineering. Or of course they might end up being super
| important 100 years later. But in the meantime
| mathematicians might be interested in them anyway.
| [deleted]
| ethanwillis wrote:
| gilbert_vanova wrote:
| > 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.
| version_five wrote:
| Reading the discussion, I think you're both right. There
| is a common sense notion of mathematics that exists
| independent of what people want to believe or agree upon.
|
| 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
| [deleted]
| vegetablepotpie wrote:
| I know nothing of OP, or the professor he talked to, but I
| think they might be talking about the higher level concepts
| of math.
|
| 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].
|
| [1] https://youtube.com/watch?v=HeQX2HjkcNo
| graycat wrote:
| The way I learned how to resolve the Russell paradox: Over
| here on the left in a pile we have the _elements_ we will
| work with. Now, for the _sets_ , they are made of the
| elements and are in a separate pile, are over here on the
| right. Sooo, with this little preliminary step, there is no
| way to consider the set of all sets that are not elements
| of themselves.
|
| 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!
| uoaei wrote:
| Math doesn't do that, models (including language: a model of
| communication around which we have already reached consensus)
| do.
| postalrat wrote:
| 2+2=4 stops being true when you use a different method of
| counting.
| ordu wrote:
| _> 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_
|
| 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.
| fspeech wrote:
| In il nome della rosa William of Baskerville tells us that
| concepts are signs and words are "signs of signs".
|
| 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.
| mola wrote:
| You got it backwards. 1984 is possible because it's
| fundemental that some truths are social constructs.
| [deleted]
| version_five wrote:
| If Winston and O'Brian both believe O'Brian is levitating,
| that still doesn't make it actually true. "Sanity is not
| statistical" as Winston says in the book
| zaphar wrote:
| I would argue that those aren't truths. They need some
| other term to describe them but truth doesn't really fit
| the bill. Conflating them with truth leads to weird
| outcomes in your reasoning.
|
| A more accurate term might be stories rather than truth.
| kenjackson wrote:
| Truth in math is not a social construct. But belief in math is
| a social construct.
| flanked-evergl wrote:
| > and that nothing was true until a social consensus had been
| reached between mathematicians.
|
| So how do calculators and computers work then?
| yazaddaruvala wrote:
| 1+1=2 is a social construct.
|
| It's simple, repeatable and therefore programable - but still
| a social construct.
| mik1998 wrote:
| It's not. It's a fact derived a priori from the definition
| of addition and the axioms of the field of real numbers.
| There's nothing special about the claim that 1 + 1 = 2. You
| can also define addition such that 1 + 1 = 0 or anything
| else.
| yazaddaruvala wrote:
| "a priori from the definition of addition" is effectively
| saying "a social construct"
|
| And I hate to break it to you but all "axioms" are social
| constructs that we use to create _hopefully useful_
| models.
| mik1998 wrote:
| No, this is not true at all. It is a certain
| _mathematical_ construction. It has nothing to do with
| society. Given the specific definition of addition in the
| field of real numbers, the statement 1 + 1 = 2 is true in
| any society.
|
| > 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 :)
| yazaddaruvala wrote:
| I'll try one last time:
|
| > 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!
| [deleted]
| jojobas wrote:
| Well if "1+1=2" is a social construct then "there are
| planets, stars, and, in general, objects" is also a social
| construct. No kind of society can make 1+1=3, and also
| there is mathematical proof that 1+1=2 (built of course on
| some sort of axioms that somebody might also consider
| social constructs).
| prmph wrote:
| Imagine a universe, where, when you combine two apples
| (or really anything), you always, through some weird
| physical process, end up with three items.
|
| Would that invalidate (or not), the statement that 1 + 1
| = 2?
| eesmith wrote:
| 1 cloud combined with 1 cloud gives 1 cloud.
| feanaro wrote:
| That's just some additional structure pertaining to,
| specifically, clouds. One cloud standing close to another
| cloud still equals two clouds, which is what 1 + 1 = 2 is
| really encoding.
| eesmith wrote:
| We know "planet" is a social construct because the
| astronomers decided, some years back, that Pluto wasn't a
| planet.
|
| 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.
| krcz wrote:
| It's a social construct in the the same sense anything not
| directly verifiable using senses is. Is there an Eiffel tower
| in Paris? Most people haven't seen it, so they can only accept
| the social consensus that it is there.
|
| 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.
| tigerlily wrote:
| To illustrate the point further, once you get to Paris how
| can you be sure it's an Eiffel tower? I guess you have to ask
| the man in the street. See the truth of it is a social
| construct. And whether you accept this as truth is a social
| construct, and so on. QED.
| quonn wrote:
| > I guess you have to ask the man in the street.
|
| 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.
| ThereIsNoWorry wrote:
| I don't know why you're being downvoted, but that is a
| perfectly valid statement.
|
| 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.
| 77pt77 wrote:
| You should read "Proofs and Refutations by Imre Lakatos" if
| you haven't already.
| q-big wrote:
| > 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.
|
| 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.
| zozbot234 wrote:
| > 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.
| eternalban wrote:
| "2 + 2 = 4"
|
| "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.
| torotonnato wrote:
| I would have asked how he could assert the truth of the
| proposition "Truth in mathematics is a social construct", since
| its truthfulness has to be a social construct too. (I assume
| that mathematics encompasses formal logic too)
| robertlagrant wrote:
| Social constructionalism is to my understanding least
| surprisingly found in universities.
|
| 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.
| 77pt77 wrote:
| What people accept as truth is a social construct.
|
| That's a different thing.
|
| Your teacher was just a sophist.
| rapsey wrote:
| Huh? Mathemathical proof is not a social construct. This makes
| no sense.
| magicalhippo wrote:
| There's a difference between a mathematical proof and
| accepted truths.
|
| 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.
| karmakurtisaani wrote:
| I suppose they mean what's commonly accepted as true and can
| be referred to as truths. No one can read all the proofs, so
| they have to trust others who have. There was that one
| example where a mathematician "proved" something terribly
| complicated using his own methods and terminology developed
| over several years. The truth value of that kind of proof is
| very much a social construct.
| Agingcoder wrote:
| Yes, mochizuki and the abc conjecture. It was an
| interesting conundrum : he was a really good mathematician,
| not a crank so his funky proof couldn't be dismissed.
| However, people were wary of approaching the proof since it
| was risky career wise (it takes time, etc). You end up with
| a weird situation where something is probably true, but you
| won't know that until a trusted group of mathematicians
| have read it and said so.
| uKVZe85V wrote:
| The mathematical truths are the fruit of the work. The social
| consensus is an implementation "detail". Yet it's the
| implementation we have. Does this make better sense now?
| kingkawn wrote:
| And yet nonetheless you feel the need to object to this
| formulation publicly and have it considered by others
| hugh-avherald wrote:
| It's certainly a social construct, but it is not _merely_ a
| social construct.
| ookdatnog wrote:
| I think you can make a philosophical argument that a fully
| formal proof, where every claim is traced all the way back to
| the axioms, is not a social construct.
|
| 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_.
| dorchadas wrote:
| > I think you can make a philosophical argument that a
| fully formal proof, where every claim is traced all the way
| back to the axioms, is not a social construct.
|
| 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.
| mpweiher wrote:
| The proof is not a social construct.
|
| 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...
| AnonCoward42 wrote:
| > The proof is not a social construct.
|
| > 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.
| TheOtherHobbes wrote:
| An apple is not an apple. An apple is a subjective
| construct that summarises the distinguishing features of
| a certain kind of object as it appears to our sense.
|
| 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.
| AnonCoward42 wrote:
| > An apple is not an apple. An apple is a subjective
| construct that summarises the distinguishing features of
| a certain kind of object as it appears to our sense.
|
| 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).
| marcusverus wrote:
| > An apple is not an apple. An apple is a subjective
| construct that summarises the distinguishing features of
| a certain kind of object as it appears to our sense.
|
| 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.
| ookdatnog wrote:
| It's not nonsense, they are exactly right.
|
| 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.
| naasking wrote:
| > So cat math likely wouldn't have integers as we know
| them, but would have some kind of size-based analogue.
|
| 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.
| gmfawcett wrote:
| IDK, it seems easy to imagine an alien mathematics based
| only upon continuous values? There's nothing obviously
| universal about discretizing things.
| naasking wrote:
| Firstly, the reals contain the integers, so there is an
| isomorphism as I said.
|
| 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.
| gmfawcett wrote:
| Eh, you're not explaining universal truths here, you're
| just anthropomorphizing. Why must it have appendages,
| organs, or populations? Why presuppose that its
| conceptual model includes particles at all? What if a
| vast, hyper-continuous intelligence simply cannot
| comprehend the concept of being discrete?
| naasking wrote:
| Firstly, those were just examples of commonly countable
| structures, even if they're not universal (which is
| debatable). Discretely countable structures are literally
| everywhere and fundamentally inescapable, which is why I
| mentioned physics. I didn't presuppose physics, the
| discrete structure of physical reality is directly
| observable, it's not some fiction we made up.
|
| 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.
| prmph wrote:
| Imagine a universe, where, when you combine two apples
| (or really anything), you always, through some weird
| physical process, ended up with three items.
|
| Would that invalidate (or not), the statement that 1 + 1
| = 2?
| mpweiher wrote:
| [Mathematical truth as a social construct]
|
| > 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...
| feanaro wrote:
| Saying Mathematics [?] Reality fails to capture a large
| portion of the story, since a subset of mathematics is
| clearly necessary to be able to encode science, and
| confirmed by science, and is in that sense a part of
| reality.
| mpweiher wrote:
| It's not necessary. It is empirically useful.
|
| 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_Effectiven
| ess...
| naasking wrote:
| > Mathematics [?] Reality
|
| Conjecture. A mathematical universe is consistent with
| everything we know, in which case math is literally the
| study of reality.
| kzz102 wrote:
| I think this is not the right way to look at it. You can think
| of mathematical proofs as computer program that is compiled by
| the mathematician by hand. There is a lot of room for error,
| but with practice and peer review, it's relatively easy to
| avoid the common errors. This human compiler also brings the
| benefit of error correcting, which commonly correct two types
| of errors: sometimes the proof makes syntactical mistakes that
| the human compiler fixes automatically, sometimes the proof
| claims something that's not fully justified (similar to calling
| a function that is not implemented), but the human compiler
| just fill in the detail themselves. The social part of
| mathematics is really about how much error the reviewer is
| willing to accept, because the reviewer can also be wrong with
| how they correct the proof.
| planck01 wrote:
| There are such reachable truths, but every mathematical system
| has a non empty set of axioms - or assumptions- which are
| 'given'.
| jojobas wrote:
| These axioms are not given in course of socialization and
| generally are observations of nature rather than human.
| roywiggins wrote:
| There are multiple set theories using different axioms. Is
| the Axiom of Choice based on an observation of nature or do
| mathematicians keep it around because it's useful? It's a
| statement about infinities that absolutely have no physical
| reality. You can do mathematics without it, and the
| question of whether to do math relying on it is a matter of
| opinion.
|
| (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)
| pencilguin wrote:
| ... of what people can and also choose to observe about
| nature.
| mpweiher wrote:
| A proof is a rhetorical device to convince others of the truth
| of a proposition.
| pencilguin wrote:
| If you can't get anybody to read your proof, does it
| demonstrate anything?
|
| 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.
| [deleted]
| IIAOPSW wrote:
| I'm going to relay a story. For a short period in my life, I
| had a roommate whom I wasn't sure if he was real for the first
| two weeks I knew him. At first it was just we agreed
| unreasonably well about our view on the world. Like I could not
| think of a single thing we differed on. But then it started to
| get uncanny. He had this way of knowing all the same trivia as
| I did. And also of not being able to recall the same bits of
| trivia I was struggling with. I'm talking really obscure sorts
| of things, not the sort of stuff you could dismiss as "20%
| random hn person recognizes it." Then there was the scavenging.
| Practically any time I mentioned off hand an idea for something
| we might have a use for, he would randomly find that or a
| similar item thrown out on the side of the curb (this was in
| NY). He wasn't buying these items, it was all just "lucky
| coincidence". Then there was the absurd situational similarity.
| We had started out as guests in an airbnb, permanent
| temporaries, but now we were both effectively bartering for
| rent making improvements on this guys apartment in exchange for
| free board. Its the sort of weird niche situation few people
| ever find themselves in, and we were both doing it.
|
| 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.
| mattigames wrote:
| "Yes profesor, truth in any academic field is result of
| agreement between the people working on that field. Just
| wondering... its everything Ok at home?"
| jules wrote:
| There are (short) computer programs where you input a
| mathematical proposition and a proof in a kind of proof
| programming language, and the program will then check if it's a
| valid proof. Saying that mathematical truth is a social
| construct is technically true but misses the point entirely.
| e12e wrote:
| How does one decide which axioms to build on?
| threatofrain wrote:
| You can choose whatever you want, and mathematicians do
| sometimes choose different axioms. The question is whether
| the _consequences_ of some axioms are up for social
| negotiation.
| auggierose wrote:
| It certainly is a social construct, because what tools are at
| my disposal to convince someone who disagrees otherwise? In
| that sense everything is a social construct.
|
| 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.
| prmph wrote:
| > 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.
|
| 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.
|
| 1. https://plato.stanford.edu/entries/intuitionism/
| auggierose wrote:
| I am very familiar with those "deep" ideas. It is really
| just about which axioms you are willing to accept. If you
| want to deny yourself the law of contradiction, that's
| fine, go ahead. There are structures where this law doesn't
| hold if you interpret the logical operators in a special
| sense, so there are valid reasons for doing so. For
| example, it is an elegant mechanism to reason inside of
| Kripke structures.
|
| 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.
| zmgsabst wrote:
| > 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.
|
| 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.
| qazxcvbnm wrote:
| > 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.
|
| 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.
| auggierose wrote:
| > Uncomputability amounts to the acknowledgement that
| though something may always be well-defined, there is no
| finite method to encompass its values
|
| That is kind of obvious, isn't it? But some people need
| more convincing than others :-)
| dorchadas wrote:
| But what this doesn't get at is that the very _system of
| logic_ we use to make proofs is a social construct. Other
| cultures have had other systems of logic, and called
| valid arguments that we wouldn 't today precisely
| _because_ they were using a different system of logic.
|
| 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.
| naasking wrote:
| > But what this doesn't get at is that the very system of
| logic we use to make proofs is a social construct. Other
| cultures have had other systems of logic, and called
| valid arguments that we wouldn't today precisely because
| they were using a different system of logic.
|
| 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.
| musingsole wrote:
| > 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.
| prmph wrote:
| Exactly the point I was making.
|
| 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.
| naasking wrote:
| Have you perchance been reading a lot of Wittgenstein?
|
| 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.
| auggierose wrote:
| > 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).
|
| Could you elaborate on that? Any references?
| naasking wrote:
| These are the implications of Godel's incompleteness
| theorems. No formal system expressive enough to encode
| arithmetic can simultaneously be both complete and prove
| its own consistency, because there will always be true
| propositions expressible in that system that cannot be
| proven in that system.
|
| 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...
| auggierose wrote:
| Ah ok, I see what you mean.
|
| This is different what I would understand under an 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.
| auggierose wrote:
| Equality is actually quite easy to axiomatise in most
| logics, here in my favourite logic:
|
| 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.
| prmph wrote:
| Can you put into words the symbolic notation you have in
| your comment? I think I understand pretty well what you
| mean, but for the avoidance of doubt, explain what the
| notation means, and then I will indicate all the
| assumptions on which it is relying.
| auggierose wrote:
| It would be somewhat lengthy to explain its meaning
| exactly here. You can read about its exact meaning and
| its context here: https://doi.org/10.47757/pal.2
|
| 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".
| auggierose wrote:
| There is a simple definition for what consistent means,
| which naasking is referring to: is it impossible to
| derive "false" purely by applying the rules of the
| logical system?
| [deleted]
| soVeryTired wrote:
| > 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.
|
| 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.
| auggierose wrote:
| Not many do this currently, that is true. But this will
| change. In a hundred years every mathematician will do
| this. I think it will reach "mainstream" much much earlier
| than this, probably around 2030.
| ChadNauseam wrote:
| Probably by 2030 and by 2040 at the latest, you will be
| able to give a proposition to a machine learning model
| and the model will output a machine-verifiable proof of
| its truth or falsity at least as often as a human can.
| aaron695 wrote:
| > that truth in mathematics was a social construct.
|
| 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.
| dist1ll wrote:
| Just because something is a social construct doesn't mean
| it's worthless, impure or required to be rejected.
|
| Going from self-reflection to nihilism is a pretty big
| overreaction.
| somat wrote:
| "That's because it's impossible for a computer to calculate
| infinite values. It can get very close to seeing a singularity,
| but it can't actually reach it"
|
| 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.
| marginalia_nu wrote:
| An ideal computer is an universal machine. A real computer has
| real limitations on what is calculable, even among the things
| that are theoretically calculable.
| moffkalast wrote:
| Sounds like we need to make a virtual universal machine and
| account for the limitations when emulating it.
| majewsky wrote:
| The problem with this is whether the emulation ever
| terminates. Simply put, you cannot emulate an infinitely
| powerful machine in finite time.
| bheadmaster wrote:
| Computers can, in fact, calculate infinite values in style of
| calculus, but they must use symbolic methods. Computer Algebra
| Systems often implement such methods.
|
| I believe this article is talking about numerical methods,
| which are always bound to finite values, because of finite
| memory.
| raverbashing wrote:
| Because infinity is a mathematical trick, to say where
| something is going but it is not calculable (it is not a number
| in itself)
| agumonkey wrote:
| well you end up in bounded axiomatic ranges
| mjburgess wrote:
| Almost all functions aren't computable, as they aren't
| discrete.
|
| A "computer" is just a function from {0,1}^N -> {0,1}^M
| MarcelOlsz wrote:
| What level of math do I need to be at to "compute" this
| comment?
| mjburgess wrote:
| Very little.
|
| 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.
| ben_w wrote:
| > function from {0,1}^N -> {0,1}^M
|
| "{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?)
|
| [0] https://en.wikipedia.org/wiki/A-Level
| Maursault wrote:
| A function is merely a rule, and rules do not _do_ anything
| other than define relationships, but this is not quite right
| either, because _who_ is defining? Taking into account Newton
| 's Second Law of Motion, which the practical application of
| cause and effect, a computer is always _always_ a person:
| _one who computes_. Consider that no matter how complicated
| they become, pencils do not calculate, cars do not drive, and
| guns do not shoot. You tell everybody. Listen to me! You 've
| gotta tell 'em! _Computers are people!_ We gotta stop them!
| Somehow!
| jojobas wrote:
| Symbolic engines are way above and beyond number crunching in
| many respects.
| mjburgess wrote:
| Mathematics is constrained by properties of abstract
| objects that the symbols are _about_.
|
| 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.
| chmod775 wrote:
| > 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.
| mjburgess wrote:
| Yes, I don't believe a concrete computer can "simulate"
| anything in the relevant sense, let alone a human brain.
|
| 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.
| e12e wrote:
| > Is it impossible to calculate infinite values in general?
|
| 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.
| fay59 wrote:
| In context, it sounds like they relied on simulations that
| don't use exact numbers. I'm guessing that they saw an IEEE-754
| floating-point infinity and then had to determine whether they
| got it because the accurate result was infinity or if the
| infinity they saw was the result of floating-point calculation
| artifacts.
| credit_guy wrote:
| > Why not?
|
| 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).
| thaumasiotes wrote:
| > Hou and Luo's work was suggestive, but not a true proof. That's
| because it's impossible for a computer to calculate infinite
| values. It can get very close to seeing a singularity, but it
| can't actually reach it -- meaning that the solution might be
| very accurate, but it's still an approximation.
|
| 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.
| macawfish wrote:
| Double precision floats have a maximum value of
| 1.7976931348623158 E + 308
| onion2k wrote:
| Aka not infinity
| PartiallyTyped wrote:
| They meant maximum finite value.
|
| 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)
| thaumasiotes wrote:
| That is not correct; they have a maximum value of positive
| infinity. See what you get when you square 1.7e+307.
| Karliss wrote:
| Just because some people decided to label floating point
| overflow conditiona as "infinity" doesn't mean it can
| actually represent values up to infinity.
|
| 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".
| gilbetron wrote:
| 1.7e+307 squared is 2.89e+614, not infinity even though a
| "computer" will say +inf
| mjburgess wrote:
| A computer is always just a function from {0,1}^N ->
| {0,1}^M
|
| 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
| thaumasiotes wrote:
| > A computer is always just a function from {0,1}^N ->
| {0,1}^M
|
| > 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?
| mjburgess wrote:
| 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.
|
| 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"
| thaumasiotes wrote:
| > 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.
|
| 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?
| mjburgess wrote:
| The issue with bit-patterns are, at least, they're
| discrete. And so cannot, eg., represent pi.
|
| 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.
| thaumasiotes wrote:
| Computers work with perfect representations of pi all the
| time. The TI-89 will do it routinely.
|
| 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.
| mjburgess wrote:
| So this is just not true, and I'm not exactly sure where
| these premises are coming from. Is it a misunderstanding
| of theoretical computer science, mathematics,
| engineering, or what?
|
| 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.
| vardump wrote:
| You can combine two doubles to double precision, to 107 bits.
|
| https://en.wikipedia.org/wiki/Quadruple-
| precision_floating-p...
| somat wrote:
| It does not have to use floating point numbers, while floating
| point numbers are hardware accelerated and thus very fast,
| there are infinite precision number libraries.
|
| 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.
| StillLrning123 wrote:
| So are they trying to find particles in the fluid with a flow
| of 0?
| Karliss wrote:
| Sure calculating infinity is easy as long as you redefine
| "infinity" to be something which isn't actual infinity. But
| it's useless for many mathematical proofs. Having overflow of
| some finite floating point calculations labelled as "infinity"
| is useful for calculations of some practical problems, but it
| shouldn't be confused with the actual mathematical concept of
| infinity just because both use the same word.
| thaumasiotes wrote:
| Floating point infinity is "actual infinity". It has all the
| correct mathematical properties. If you want to slam a
| special IEEE constant, you should slam 0, which has different
| properties from mathematical zero.
| majewsky wrote:
| According to my Javascript console (using IEEE754 double
| precision), 1e308+1e308 equals Infinity. That's not "actual
| infinity".
| photochemsyn wrote:
| > "Yet much remains unknown about the Euler equations --
| including whether they're always an accurate model of ideal fluid
| flow."
|
| 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."
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