Struggles with the Continuum(arxiv.org)
arxiv.org
Struggles with the Continuum
https://arxiv.org/abs/1609.01421
8 comments
Let me offer an alternative motivation. Suppose that the universe is ultimately discrete at its smallest scales, bounded by some finite length, and everything advances in discrete 'ticks' of time. There would be no physical realisation of the real numbers.
But they would still be _useful_ as a computational device. Calculus seems intimidating and hard at first glance, but it's often easier than working with the discrete problem. If you've ever taken a probability course you may have encountered this: the binomial distribution for a large number of trials can be approximate by a Gaussian (at least, if I'm not looking too far into its tails). At a more advanced level, quants model the price process of securities as stochastic differential equations. Of course, this is nonsense - prices are in discrete number of pennies, for instance. But the error of this approximation is tolerable to produce an analytic result (the Black-Scholes formula), just so long as one remembers it isn't an exact result.
If you want a more physical example, consider the fact that we treat fluids as continua, modelled by the Navier-Stokes equations. Of course they actually consist of discrete particles, but so long as our scales of interest are greater than the mean-free-path-length, a continuous approximation is far more practical for calculations than modelling a huge number of discrete particles.
But they would still be _useful_ as a computational device. Calculus seems intimidating and hard at first glance, but it's often easier than working with the discrete problem. If you've ever taken a probability course you may have encountered this: the binomial distribution for a large number of trials can be approximate by a Gaussian (at least, if I'm not looking too far into its tails). At a more advanced level, quants model the price process of securities as stochastic differential equations. Of course, this is nonsense - prices are in discrete number of pennies, for instance. But the error of this approximation is tolerable to produce an analytic result (the Black-Scholes formula), just so long as one remembers it isn't an exact result.
If you want a more physical example, consider the fact that we treat fluids as continua, modelled by the Navier-Stokes equations. Of course they actually consist of discrete particles, but so long as our scales of interest are greater than the mean-free-path-length, a continuous approximation is far more practical for calculations than modelling a huge number of discrete particles.
> Suppose that the universe is ultimately discrete at its smallest scales, bounded by some finite length, and everything advances in discrete 'ticks' of time. There would be no physical realisation of the real numbers.
I'm not a mathematician, or a physicist for that matter - but is that true? You've precluded "infinitely small". But you didn't exclude "infinitely big". If the universe contains infinitely big things, then to it anything finite sized is infinitely small, isn't it? To finish off the thought bubble, as far as I know the universe may end up being infinitely big.
I'm not a mathematician, or a physicist for that matter - but is that true? You've precluded "infinitely small". But you didn't exclude "infinitely big". If the universe contains infinitely big things, then to it anything finite sized is infinitely small, isn't it? To finish off the thought bubble, as far as I know the universe may end up being infinitely big.
Bring us another complete ordered field and I’ll pay handsomely for it.
Professional mathematicians have given it a lot of thought. Your tutor probably just didn't wonder enough about it. I did my graduate studies in numerical methods of mathematical physics, and everyone I talked to agreed, that the continuum assumption creates a lot of problems. But even so, the nice things they give in the space of functions and numerical convergence has been more important than the difficulties they cause. And hence, continuum analysis it is.
In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.
In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.
> In one of my papers, I actually assumed that two features in material have to be minimum epsilon distance away to show that the method convergences in a reasonable amount of computational effort. (where epsilon is arbitrarily small) So, I did break the continuum barrier to do some useful physics ones.
Forgive me for asking the obvious question, but isn't that… just "atoms"?
(And now I remember my excitement when I first got to play with satellite data and python, did a Fourier transform to see if there was any temporal pattern to the environment, and was disappointed to realise I'd just rediscovered "winter and summer" in quite possibly the most ridiculous way possible).
Forgive me for asking the obvious question, but isn't that… just "atoms"?
(And now I remember my excitement when I first got to play with satellite data and python, did a Fourier transform to see if there was any temporal pattern to the environment, and was disappointed to realise I'd just rediscovered "winter and summer" in quite possibly the most ridiculous way possible).
Some have given it a lot of thought, but not nearly as many as I would have expected. Despite such questions being philosophical in nature it seems to mostly be mathematical physicists and applied mathematicians who spend time on it. Pure mathematicians mostly don’t care; they’re not interested in existence or ‘reality’ — if there’s complexity in an idea it’s worth studying whether it has anything to say about models of the universe or not.
I guess what I’m trying to say is just that I’ve always been surprised by the lack of interest most pure mathematicians show towards questions of ‘what exists’ or ‘what the purpose of mathematics is’ and so on. I would have expected such (clearly) intellectually curious people to be less myopic.
Because the world is a big place, one can find countless essays on these sorts of topics. But it’s still a minority sport. John Baez is a brilliant outlier in many ways.
I guess what I’m trying to say is just that I’ve always been surprised by the lack of interest most pure mathematicians show towards questions of ‘what exists’ or ‘what the purpose of mathematics is’ and so on. I would have expected such (clearly) intellectually curious people to be less myopic.
Because the world is a big place, one can find countless essays on these sorts of topics. But it’s still a minority sport. John Baez is a brilliant outlier in many ways.
I agree that pure mathematicians might not care about applications. That is simply not their job.
But successful applications of the real numbers are everywhere present in physics, engineering, statistics, and the other natural sciences. The funny thing is that many of those applications don't care either if real numbers represent an aspect of reality. They are a well-enough approximation, and that makes them work.
But successful applications of the real numbers are everywhere present in physics, engineering, statistics, and the other natural sciences. The funny thing is that many of those applications don't care either if real numbers represent an aspect of reality. They are a well-enough approximation, and that makes them work.
The job of an academic is, one could say, to wonder about things (and then pursue them with scholarly rigour). So it seems strange to me that academics think of themselves as having such narrowly-defined roles even within their own subject. But I guess the (pragmatic) answer is that to be successful you must keep focused.
Of course, this is opening another whole can of worms that is perhaps starting to become off-topic.
Of course, this is opening another whole can of worms that is perhaps starting to become off-topic.
As you succinctly said it yourself, pure mathematics is indeed by definition unconcerned about whether their objects of study actually "exist". That's the business of physicists and certain subfields of philosophy.
Pure mathematics has indeed been challenged whether it actually makes sense to further fund it, but occasionally their objects of study turn out to be useful. For example number theory, which is the foundation of modern asymmetric cryptography.
But even if their research never turns out to be useful, pure mathematicians are unconcerned about it since they don't approach their research objects with any assumption regarding that.
Pure mathematics has indeed been challenged whether it actually makes sense to further fund it, but occasionally their objects of study turn out to be useful. For example number theory, which is the foundation of modern asymmetric cryptography.
But even if their research never turns out to be useful, pure mathematicians are unconcerned about it since they don't approach their research objects with any assumption regarding that.
Because it does not generate very interesting models and becomes extremely and intractably difficulty really fast.
You could write your problem as a discrete system and often that is kind of it. People have tried for a long time and the solution methods to the problems are almost impossible or too expensive.
You could write your problem as a discrete system and often that is kind of it. People have tried for a long time and the solution methods to the problems are almost impossible or too expensive.
Do the same with three or more attractors or exit basins and you will find that even with a 0 or negative Laponov exponent (non-chaotic), some systems are indeterminate.
While typically taught as chaotic, Newton's fractal is probably the most accessible example.
No matter how small your epsilon is, your piece of the continua will either contain one root or all roots.
It is an indecomposable continua.
Predator pray with fear and refuge is an example where you hit this in numerical systems.
While typically taught as chaotic, Newton's fractal is probably the most accessible example.
No matter how small your epsilon is, your piece of the continua will either contain one root or all roots.
It is an indecomposable continua.
Predator pray with fear and refuge is an example where you hit this in numerical systems.
I mean, real numbers certainly do exist mathematically. They're not very difficult to rigorously construct, so I can see why your tutor would be bewildered by the statement.
Whether or not they're a good description of coordinates in spacetime is a pretty separate matter.
Whether or not they're a good description of coordinates in spacetime is a pretty separate matter.
I know how they can be constructed. But mathematical ideas don’t exist in a vacuum — they’re supposed to reflect some intuition or describe some (possibly very abstract) reality. My worry came from an inability to reconcile the theory with any such intuition.
You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.
You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.
> they’re supposed to reflect some intuition or describe some (possibly very abstract) reality.
Says who? Math is the study of finding all possible logical conclusions from a given set of assumptions. Reality has nothing to do with it.
Says who? Math is the study of finding all possible logical conclusions from a given set of assumptions. Reality has nothing to do with it.
> all possible logical conclusions
I don’t think this is accurate. All logical conclusions would be a horrible mess of symbols because we don’t care about most of them (because they’re considered trivial) and the list would be infinite and filled with statements that are very similar to each other. The subject isn’t about brute force in that way.
It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. What those two words mean is then the interesting question (and, of course, a matter of personal taste). Reality certainly has something to do with that choice, which is why I don’t think you can escape it. It’s fundamental to the subject, no matter how seemingly abstract you get.
I don’t think this is accurate. All logical conclusions would be a horrible mess of symbols because we don’t care about most of them (because they’re considered trivial) and the list would be infinite and filled with statements that are very similar to each other. The subject isn’t about brute force in that way.
It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. What those two words mean is then the interesting question (and, of course, a matter of personal taste). Reality certainly has something to do with that choice, which is why I don’t think you can escape it. It’s fundamental to the subject, no matter how seemingly abstract you get.
> It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions.
But that depends on what particular people find meaningful or interesting. Lots of people--or at least lots of mathematicians :)--appear to find many things interesting that have nothing at all to do with "reality".
Also, it's important not to confuse "reality" with our models of it. Real numbers are unquestionably very useful in constructing models of reality. That will still be true even if it turns out that actual "reality" is not a continuum.
But that depends on what particular people find meaningful or interesting. Lots of people--or at least lots of mathematicians :)--appear to find many things interesting that have nothing at all to do with "reality".
Also, it's important not to confuse "reality" with our models of it. Real numbers are unquestionably very useful in constructing models of reality. That will still be true even if it turns out that actual "reality" is not a continuum.
> But that depends on what particular people find meaningful or interesting
Yes. That’s why I said it’s a matter of taste.
My claim is that what makes things meaningful is how they relate to reality, and by reality I mean things like the three dimensional world and the passing of time we perceive. Whilst most of pure mathematics seems ridiculously abstract on first glance, I don’t believe there is a single topic that truly has no connection to reality.
> it's important not to confuse "reality" with our models of it
This is a frequently repeated point and I’m inclined to agree, but any such model also exists in reality and so it’s hard to know where to draw the line.
Yes. That’s why I said it’s a matter of taste.
My claim is that what makes things meaningful is how they relate to reality, and by reality I mean things like the three dimensional world and the passing of time we perceive. Whilst most of pure mathematics seems ridiculously abstract on first glance, I don’t believe there is a single topic that truly has no connection to reality.
> it's important not to confuse "reality" with our models of it
This is a frequently repeated point and I’m inclined to agree, but any such model also exists in reality and so it’s hard to know where to draw the line.
> any such model also exists in reality
Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city.
Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.
Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city.
Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.
Of course there’s a distinction between a map and the thing it’s a map of. But they’re both parts of reality.
It's also perfectly possible to draw maps of places that don't exist in reality, i.e., that only exist in somebody's brain.
A good point. So the real numbers exist in the same sense that anything else they might describe does.
My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.
My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.
Yes, I agree. Real numbers are a quite nebulous construct for which the need is a bit hard to see. Complex numbers are basically a hack to solve quadratic equations in all cases.
I agree with what you have said but have an additional worry. It seems to me that the standard logic we use might be heavily rooted in the real world. It either rains or doesn't rain seems to closely reflect how the classical world works. But would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain?
> would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain?
You don't have to change the laws of logic that you use to do unitary QM (which, taken to extremes, is what gives you a "many worlds universe"). You can describe it just fine using mathematical equations based on the same laws of logic you're familiar with.
In other words, what you are calling "standard logic" actually contains many more possibilities than just the simple Boolean "rains or doesn't rain".
You don't have to change the laws of logic that you use to do unitary QM (which, taken to extremes, is what gives you a "many worlds universe"). You can describe it just fine using mathematical equations based on the same laws of logic you're familiar with.
In other words, what you are calling "standard logic" actually contains many more possibilities than just the simple Boolean "rains or doesn't rain".
Sure, if the system is expressive enough, you can emulate other systems in it, you can build fuzzy logic on top of classical logic. Many worlds was just a random example which is not too far from the classical world. But can we not imagine an universe that is very different from ours, that outright contradicts our logic? After all logic is just build on top of axioms that make sense to us, I don't see that this necessarily grants them some universal truth.
> can we not imagine an universe that is very different from ours, that outright contradicts our logic?
Can you give an example?
If you can't, doesn't that in itself answer your question?
Can you give an example?
If you can't, doesn't that in itself answer your question?
Not necessarily. How and what we can think might be constraint by our brains or maybe even by the universe itself. I don't think any human can make sense of two time dimensions but it seem premature to me to conclude that no universe could have more than one time dimension. Anything I could suggest would by definition not make any sense to us as it defies our logic but could it really not exist? It might of course just be a stupid idea. How would you even think about something like this if you throw out the foundation of our thinking?
> I don't think any human can make sense of two time dimensions
Sure humans can; Greg Egan built a whole sci-fi universe that way:
https://www.gregegan.net/DICHRONAUTS/00/DPDM.html
But two time dimensions doesn't contradict our logic. (It contradicts our best current theories of physics, but that's not the same thing as contradicting our logic. Egan's model is logically consistent.) So this is not an example of what you were claiming.
> Anything I could suggest would by definition not make any sense to us
In other words, you are saying humans can't imagine a universe that contradicts our logic. I agree. And that is a "no" answer to the question you posed in the GGP post to this one.
Whether universes that contradict our logic could "exist", even if we can't imagine them, is a different question from the one you asked in that post. As for that:
> How would you even think about something like this if you throw out the foundation of our thinking?
Indeed. And if we can't even consistently think about such a thing, what's the point?
Sure humans can; Greg Egan built a whole sci-fi universe that way:
https://www.gregegan.net/DICHRONAUTS/00/DPDM.html
But two time dimensions doesn't contradict our logic. (It contradicts our best current theories of physics, but that's not the same thing as contradicting our logic. Egan's model is logically consistent.) So this is not an example of what you were claiming.
> Anything I could suggest would by definition not make any sense to us
In other words, you are saying humans can't imagine a universe that contradicts our logic. I agree. And that is a "no" answer to the question you posed in the GGP post to this one.
Whether universes that contradict our logic could "exist", even if we can't imagine them, is a different question from the one you asked in that post. As for that:
> How would you even think about something like this if you throw out the foundation of our thinking?
Indeed. And if we can't even consistently think about such a thing, what's the point?
Such systems of logic are quite useful for dealing with our own reality as well. Paraconsistent logics disallow or restrict some of the axioms of classical logics to avoid some contradictions. In turn, it loses completeness to some degree. It has similarities to intuitionistic logic, but intuitionistic logic has other aims.
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The subject you’re looking for is philosophy of mathematics. Here’s a good place to start [1]. Unfortunately, if you’re looking for certainty you won’t find it there. Philosophy of mathematics is fascinating but studying it will give you more questions but few answers. One advantage of studying that I will vouch for is that you’ll learn that most of the arguments you had were already thought of many years ago.
[1] https://plato.stanford.edu/entries/philosophy-mathematics/
[1] https://plato.stanford.edu/entries/philosophy-mathematics/
Yes! I am aware of PoM, just surprised so many mathematicians are seemingly disinterested.
This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...
This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...
I think most mathematicians are uninterested in PoM because their interest in math comes from an aesthetic impulse and a problem-solving drive. In short, they like math because it’s beautiful and challenging, and challenging problems are fun for them to solve.
They may also be motivated by prestige, to varying degrees, though I think mathematicians are fairly low on the prestige-valuation scale (viz. Grigori Perelman).
Though I enjoy PoM myself, it actually feels a bit weird to wonder why mathematicians don’t care about it. To me, it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare.
They may also be motivated by prestige, to varying degrees, though I think mathematicians are fairly low on the prestige-valuation scale (viz. Grigori Perelman).
Though I enjoy PoM myself, it actually feels a bit weird to wonder why mathematicians don’t care about it. To me, it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare.
> it actually feels a bit weird
I won’t argue with that. :-)
> it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare
You’re exactly right. That is why I will never be a chess grandmaster!
I won’t argue with that. :-)
> it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare
You’re exactly right. That is why I will never be a chess grandmaster!
But haven't real numbers served us pretty well in application as well, for most of the time?
Modern physics is anything but intuitive in my opinion. Maybe reality isn't well described by real numbers, or maybe it is; but my sense is the reason why physicists have used real numbers is because they've worked pretty well up until a certain relatively recent point, when other intuitions about reality have started to fall apart as well.
Modern physics is anything but intuitive in my opinion. Maybe reality isn't well described by real numbers, or maybe it is; but my sense is the reason why physicists have used real numbers is because they've worked pretty well up until a certain relatively recent point, when other intuitions about reality have started to fall apart as well.
Part of the problem is that they maybe work too well. A lot of particle theory is model fitting without probiding us new insights into a possible underlying reality. Or something that would force a paradigm shift. Real evidence of something completely contradictive to our existing theories would be a dream come true.
Mathematicians pay a lot of attention to real numbers simply because ones runs into them everywhere. If rational numbers were enough to make analysis, calculus, and ultimately physics work out, we would certainly stick with those simpler objects. Real numbers are strange fellas for sure, but nature being describable by rational numbers alone ought to be testable.
Continuous math is just a finite set of symbols that obey a set of laws (maybe a finite set of laws, or maybe an infinite set of laws, formulated in a higher level logic).
Even if you don't believe that there exist anything that you observe and explore, real analysis can be still a fun game to play.
Even if you don't believe that there exist anything that you observe and explore, real analysis can be still a fun game to play.
Isn't that part of the problem? Mathematicians world separate out things that apply to the real world vs things that for now are just fun games they like to play (but maybe in the future will be found to apply).
Mathematicians don't "believe" in real numbers. They're part of an abstract logical framework, not a statement about the nature of reality. Mathematical objects arise from definitions, not measurements and observations.
"Mathematical objects arise from definitions, not measurements and observations."
Abstract logical frameworks are not pulled from thin air. Human brains develop such frameworks with a physical brain that receives sensory input from a physical world, and hence there is a deep connection between the physical world and the abstract logical frameworks in many ways.
Abstract logical frameworks are not pulled from thin air. Human brains develop such frameworks with a physical brain that receives sensory input from a physical world, and hence there is a deep connection between the physical world and the abstract logical frameworks in many ways.
Definitions are created to be useful or interesting or both, and sometimes those specifiers overlap with physical reality. But that doesn't mean they're created to match reality or to tell you anything about it.
I remember struggling as an undergrad to come up with a metaphor for visualizing what a group homomorphism is, aiming to develop better intuition when working with them. It's hard because the real world doesn't contain any, and all the "it's kind of like X" types of examples you can think of aren't really useful when mapped back to the mathematical domain. If math described the real world, I'd expect this type of thing to be much easier.
I remember struggling as an undergrad to come up with a metaphor for visualizing what a group homomorphism is, aiming to develop better intuition when working with them. It's hard because the real world doesn't contain any, and all the "it's kind of like X" types of examples you can think of aren't really useful when mapped back to the mathematical domain. If math described the real world, I'd expect this type of thing to be much easier.
An example of a group homomorphism in physics is the relationship between the rotational symmetry group of a physical system (often the group SO(3) for three-dimensional rotations or SU(2) for spin systems) and the representation of that group by the angular momentum operators in quantum mechanics. The angular momentum operators in quantum mechanics form a representation of the symmetry group because they satisfy the same commutation relations as the infinitesimal generators of the group. This means that the algebraic structure of the rotational symmetry group is mirrored by the algebraic structure of the angular momentum operators, and the way these operators transform states in the Hilbert space of the quantum system reflects the symmetry properties of the system.
I'm not sure if you intend this as a helpful analogy for an undergrad or as an argument in favor of groups/homomorphisms being created to match reality, but I don't think it applies very well to either.
(Though it is very interesting!)
(Though it is very interesting!)
An important part of mathematics is indeed formalizing such objects. Mathematicians go further, question these assumptions, and abstract from them. For example, removing constraints on fields gives us rings, groups, semigroups, and categories, and exploring what can still be said about them.
But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets. The only restriction is that they should not lead to immediately obvious contradictions, since those would probably be kind of boring!
But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets. The only restriction is that they should not lead to immediately obvious contradictions, since those would probably be kind of boring!
"But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets."
I do agree, but that we don't do that in reality is exactly my point.
I do agree, but that we don't do that in reality is exactly my point.
The typical working mathematician is a Platonist on weekdays and a formalist on Sundays
R. Hersh, 1979
R. Hersh, 1979
I'm TAing real analysis 1 this semester, and that course ought to give an answers to this. To name a few:
1. Every Cauchy-sequence converges, i.e. every sequence that ought to converge does so. This implies the existence of roots, pi, e, sine and cosine functions defined by series representations.
2. The real numbers are the only archimedean ordered field which satisfies point 1.
3. If f is a continuous function on the interval [0,1] with f(0) < 0, f(1) > 0, then f has a zero in [0,1]. "If a function which can be drawn in one stroke crosses the x-Axis, at has a zero at the intersection point".
As derivatives and integrals are typically defined using limits, having good results about the existence of limits is important theory wise.
That said, I guess you could built up most of analysis 1 around the concept of a computable number, restricting yourself to computable functions. But I'm not sure that would make the subject easier at all.
1. Every Cauchy-sequence converges, i.e. every sequence that ought to converge does so. This implies the existence of roots, pi, e, sine and cosine functions defined by series representations.
2. The real numbers are the only archimedean ordered field which satisfies point 1.
3. If f is a continuous function on the interval [0,1] with f(0) < 0, f(1) > 0, then f has a zero in [0,1]. "If a function which can be drawn in one stroke crosses the x-Axis, at has a zero at the intersection point".
As derivatives and integrals are typically defined using limits, having good results about the existence of limits is important theory wise.
That said, I guess you could built up most of analysis 1 around the concept of a computable number, restricting yourself to computable functions. But I'm not sure that would make the subject easier at all.
You probably mean f(1) > 0.
Any number that cannot be effectively computed does not exist; there are actual hard physical limits to Numbers.
I disagree with your statement and provide Graham's number as a counterexample.
https://en.m.wikipedia.org/wiki/Graham%27s_number (edit: and TREE(3) )
https://en.m.wikipedia.org/wiki/Graham%27s_number (edit: and TREE(3) )
One of may teacher of Calculus/Analysis (in Italy we say Analisi) used to say that in the real world there are only rational numbers. The mathematicians had to invente the Real numbers in order to perform analysis
I heard a lot about questioning the "existence" of real numbers and if they are a realistic description of reality, but I kinda feel like a portion of that uncertanty should also be applied to integers as well. How does the notion of a discrete element relate to our current world-view of fields and waves? Can you truly have a "single thing", isolated from its externals, constituting a discrete element? What would that be? What can you put in a box and not have it influence the would outside (or the box itself as well), thus extending its reach and making it a part of a greater whole?
For what it's worth, I agree. I can't get myself to care about any math that cares about the detailed properties of reals because I care about understanding reality with math and that can't be relevant.
It seems to me that it's not so much that physics uses the rationals, though, as that it uses something like "fuzzy reals", the reals up to a certain resolution. IMO it should be impossible for physics to ever use any sort of infinitely-precise property of the number system it's using, simply because the number system is arbitrary.
The argument for this (besides "it's obvious") is that the only ways physics can ever make numbers is (a) counting things, (b) measuring things with something that acts like a yardstick, or (c) apply formulas to numbers you already have. The first only creates integers. The second only creates fuzzy real numbers up to the resolution of your measurement. The third propagates the discreteness of the first and the uncertainty of the second through whatever calculation you do. Hence, physics cannot deal with infinitely precise descriptions of reality.
One might argue that, once you have a model, you can get infinitely precise within the model (e.g. computing infinitely rich Feynman diagrams in QFT). I have no problem with that. I just don't believe the precise differentiability properties of physical objects like fields should be allowed to be used in the model, because they seem unphysical.
It seems to me that it's not so much that physics uses the rationals, though, as that it uses something like "fuzzy reals", the reals up to a certain resolution. IMO it should be impossible for physics to ever use any sort of infinitely-precise property of the number system it's using, simply because the number system is arbitrary.
The argument for this (besides "it's obvious") is that the only ways physics can ever make numbers is (a) counting things, (b) measuring things with something that acts like a yardstick, or (c) apply formulas to numbers you already have. The first only creates integers. The second only creates fuzzy real numbers up to the resolution of your measurement. The third propagates the discreteness of the first and the uncertainty of the second through whatever calculation you do. Hence, physics cannot deal with infinitely precise descriptions of reality.
One might argue that, once you have a model, you can get infinitely precise within the model (e.g. computing infinitely rich Feynman diagrams in QFT). I have no problem with that. I just don't believe the precise differentiability properties of physical objects like fields should be allowed to be used in the model, because they seem unphysical.
There's likely some work in the direction of numerical analysis, computable analysis, and formal logic -- all of which tries to make numerical programming more rigorous -- which might also address some of the philosophical unease people have about the real numbers.
In computable analysis, the real numbers are there, and we may compute directly on them using a machine model. In this framework, all computable functions are continuous, and conversely, nearly all continuous functions encountered in practice are computable. It's neat, but exposure to numerical analysis reveals two holes.
First hole is: The time and memory costs of such a theory are infeasible. You can try to fix it by replacing reals with rationals, but rationals are also not feasible for similar reasons. Ultimately, we are led to the floating point numbers, which allow us to rigorously control time and memory usage. We can attempt to make floats more rigorous by replacing them with intervals, but intervals are impractical for a different set of reasons.
Second hole is: People routinely compute things which are uncomputable. For instance, the Singular Value Decomposition in linear algebra is uncomputable, while also being a staple of numerical computing. This is clearly a problem for the definition of the term "uncomputable". When diagnosing the problem, you can see that in numerical linear algebra, there are three relevant notions: Forward error, backward error, and condition number. The example of SVD is "uncomputable" because it is impossible to devise an algorithm which has arbitrarily low forward error. But we can devise an algorithm with arbitrarily low backward error and condition numbers for the output matrices. If you consider instead computing the Jordan Normal Form, then this is in some sense "more uncomputable" than SVD, since any algorithm which guarantees a low backward error is forced to output high condition-numbered matrices; and vice versa.
In computable analysis, the real numbers are there, and we may compute directly on them using a machine model. In this framework, all computable functions are continuous, and conversely, nearly all continuous functions encountered in practice are computable. It's neat, but exposure to numerical analysis reveals two holes.
First hole is: The time and memory costs of such a theory are infeasible. You can try to fix it by replacing reals with rationals, but rationals are also not feasible for similar reasons. Ultimately, we are led to the floating point numbers, which allow us to rigorously control time and memory usage. We can attempt to make floats more rigorous by replacing them with intervals, but intervals are impractical for a different set of reasons.
Second hole is: People routinely compute things which are uncomputable. For instance, the Singular Value Decomposition in linear algebra is uncomputable, while also being a staple of numerical computing. This is clearly a problem for the definition of the term "uncomputable". When diagnosing the problem, you can see that in numerical linear algebra, there are three relevant notions: Forward error, backward error, and condition number. The example of SVD is "uncomputable" because it is impossible to devise an algorithm which has arbitrarily low forward error. But we can devise an algorithm with arbitrarily low backward error and condition numbers for the output matrices. If you consider instead computing the Jordan Normal Form, then this is in some sense "more uncomputable" than SVD, since any algorithm which guarantees a low backward error is forced to output high condition-numbered matrices; and vice versa.
You can use spigot as a practical implementation of exact/constructive real numbers. It's in the Debian software repo, or you can find it at https://www.chiark.greenend.org.uk/~sgtatham/spigot/
The main weirdness about exact reals is that "trivial" computations like sin(asin(0.12345)) will actually hang after outputing 0.1234, since spigot can't know whether the result is 0.12344999..99something or 0.12345000..00something; the "something" part that might break the tie never appears. There are cases where an 'exact' computation can't even decide whether the result is exactly zero or some number other than zero.
The main weirdness about exact reals is that "trivial" computations like sin(asin(0.12345)) will actually hang after outputing 0.1234, since spigot can't know whether the result is 0.12344999..99something or 0.12345000..00something; the "something" part that might break the tie never appears. There are cases where an 'exact' computation can't even decide whether the result is exactly zero or some number other than zero.
> It seems that most professional mathematicians have never really given it much thought
This statement cannot be more wrong. See continuum hypothesis for example.
In the example you gave between you and your tutor, you represent someone who is a complete outsider or amateur, while the tutor represents someone (typical for people hold an undergraduate degree) who thinks what there are taught are the final words describing the truth. None of the 2 samples are in the “professional” category. Why would you think from this you can draw conclusion about “professional mathematician”?
And I don’t think “professional mathematician” is the right terminology here, but I’m not a “professional linguist” so I’m not certain here.
The paper you are commenting on is in mathematical physics. Mathematics rigorously defines self consistent systems, while (mathematical) physics concerns which “system” or language is describing our universe. So I’d say you entirely missed the point. (By the way, such speculation is not new in mathematical physics either, so don’t think “professional physicists” never given it much thought either.)
I’m surprised to see the top comment here missing the point so much. May be this is really just hackers news where topics outside the hacker circle is mostly noise rather than signal.
This statement cannot be more wrong. See continuum hypothesis for example.
In the example you gave between you and your tutor, you represent someone who is a complete outsider or amateur, while the tutor represents someone (typical for people hold an undergraduate degree) who thinks what there are taught are the final words describing the truth. None of the 2 samples are in the “professional” category. Why would you think from this you can draw conclusion about “professional mathematician”?
And I don’t think “professional mathematician” is the right terminology here, but I’m not a “professional linguist” so I’m not certain here.
The paper you are commenting on is in mathematical physics. Mathematics rigorously defines self consistent systems, while (mathematical) physics concerns which “system” or language is describing our universe. So I’d say you entirely missed the point. (By the way, such speculation is not new in mathematical physics either, so don’t think “professional physicists” never given it much thought either.)
I’m surprised to see the top comment here missing the point so much. May be this is really just hackers news where topics outside the hacker circle is mostly noise rather than signal.
> This statement cannot be more wrong. See continuum hypothesis for example.
1) ‘it seems that no mathematician has ever considered this’ would be even more (and certainly is) wrong, so my statement certainly could be more wrong (if indeed you think it is).
2) The absolute vast majority of mathematicians would struggle to even state the CH. If you’ve personally thought a lot about set theory, I can see why you might view the world otherwise.
About ‘professional mathematicians’, I too hesitated before writing that. Perhaps it is the wrong term. What I intended by it was ‘people who do (pure) mathematics for a living’, so basically mathematicians who work at a university and primarily carry out research. ‘Mathematicians’ would not suffice, since that’s an even more vague term used in all sorts of completely different ways. Please suggest a better term if you have one.
> Why would you think from this you can draw conclusion about “professional mathematician”?
I don’t. My conclusions are drawn from the sample I have, which includes quite a variety of mathematicians I’ve chatted with online and in person. In case it’s the word ‘tutor’ that confused you, it’s worth mentioning that the person in question is a renowned mathematician (and professor, which means something in the UK, unlike in the US) who has been doing research for about four decades. If that’s not ‘pro’ then I don’t know what is.
My comment doesn’t directly address the article and isn’t physics-related so I’m sorry if that means I’ve ‘missed the point’. I was just adding something I thought might be of interest to others, and it seems it was.
1) ‘it seems that no mathematician has ever considered this’ would be even more (and certainly is) wrong, so my statement certainly could be more wrong (if indeed you think it is).
2) The absolute vast majority of mathematicians would struggle to even state the CH. If you’ve personally thought a lot about set theory, I can see why you might view the world otherwise.
About ‘professional mathematicians’, I too hesitated before writing that. Perhaps it is the wrong term. What I intended by it was ‘people who do (pure) mathematics for a living’, so basically mathematicians who work at a university and primarily carry out research. ‘Mathematicians’ would not suffice, since that’s an even more vague term used in all sorts of completely different ways. Please suggest a better term if you have one.
> Why would you think from this you can draw conclusion about “professional mathematician”?
I don’t. My conclusions are drawn from the sample I have, which includes quite a variety of mathematicians I’ve chatted with online and in person. In case it’s the word ‘tutor’ that confused you, it’s worth mentioning that the person in question is a renowned mathematician (and professor, which means something in the UK, unlike in the US) who has been doing research for about four decades. If that’s not ‘pro’ then I don’t know what is.
My comment doesn’t directly address the article and isn’t physics-related so I’m sorry if that means I’ve ‘missed the point’. I was just adding something I thought might be of interest to others, and it seems it was.
Ok, fair enough.
But shouldn't a direct comment be addressing to the paper instead? I might be wrong and am certainly not well versed in HN.
I am not disappointed in your comment per se, I'm disappointed that is the top comment. May be because replies to your comment got more upvote (again, not knowing HN well enough to understand this.) But if that's the case, that's more because of a controversial comment attracting more refutals than the comment itself being more relevant to the discussion.
But shouldn't a direct comment be addressing to the paper instead? I might be wrong and am certainly not well versed in HN.
I am not disappointed in your comment per se, I'm disappointed that is the top comment. May be because replies to your comment got more upvote (again, not knowing HN well enough to understand this.) But if that's the case, that's more because of a controversial comment attracting more refutals than the comment itself being more relevant to the discussion.
> shouldn't a direct comment be addressing to the paper instead?
Perhaps, yes. I’m not sure what’s considered acceptable and it’s not as if I intended it to reach the ‘top’ (if it has?).
Perhaps, yes. I’m not sure what’s considered acceptable and it’s not as if I intended it to reach the ‘top’ (if it has?).
No hard feelings. It tells more about the community than you. Cheers.
I find the idea of a universe with a discrete substrate/structure both natural and compelling. I wish we’d explore it more. If you really think about it, it’s hard to imagine nature would just happen into a system with irrational numbers as we naïvely (undergraduate level college and below) understand them. Things clearly look infinite on a macro level, but “what type of discrete systems give rise to the observed macro behavior” is a field that needs more exploration.
The maths analog is alternative number systems, which at least seem to be gaining interest and popularity.
The metaphor for programmers: is the universe just a beautifully seeded Conway’s game of life?
The maths analog is alternative number systems, which at least seem to be gaining interest and popularity.
The metaphor for programmers: is the universe just a beautifully seeded Conway’s game of life?
The real numbers are irrational almost everywhere, they are uncomputable 'almost everywhere' also.
“what type of discrete systems give rise to the observed macro behavior” is what we have been doing forever through Laplacian determinism.
There are no alternative numbers systems that get around this.
That is the whole point of the problem of the cardinality of the continuum.
We know that Laplacian determinism fails, with quantum mechanics being the easiest counterexample but Cantor diagonalization is another.
The sofar unsuccessful efforts to produce string theory is an example of this effort, changing world lines into sheets.
With the discovery of strange non chaotic attractors and the discovery that our numerical systems are subject to the WadA property in time delayed and Hamilton systems it is possible that a unified model may not exist.
None of this is undergrad though, due to how we have chosen to teach it.
But it is very possible that existential quantifiers are what math limits is to building models of.
“what type of discrete systems give rise to the observed macro behavior” is what we have been doing forever through Laplacian determinism.
There are no alternative numbers systems that get around this.
That is the whole point of the problem of the cardinality of the continuum.
We know that Laplacian determinism fails, with quantum mechanics being the easiest counterexample but Cantor diagonalization is another.
The sofar unsuccessful efforts to produce string theory is an example of this effort, changing world lines into sheets.
With the discovery of strange non chaotic attractors and the discovery that our numerical systems are subject to the WadA property in time delayed and Hamilton systems it is possible that a unified model may not exist.
None of this is undergrad though, due to how we have chosen to teach it.
But it is very possible that existential quantifiers are what math limits is to building models of.
Couldn't agree more, though when I see this brought up, I've always seen "people who seem to know more than me" saying that it just leads to other problems, and they kind of hand-wave it away... but I still think there should be some more serious exploration of this. How could "real" numbers _truly_ exist, any more than "imaginary" ones?
I think anybody who thinks real numbers exist will also say that imaginary numbers exist. "real" and "imaginary" in the context of numbers are just arbitrary labels assigned for historical reasons, they're not meant to convey any philosophical judgements about existence.
I don't think so. You can have imaginary rationals, or even imaginary natural numbers (that is, a+bi, where both a and b are rational numbers or integers).
Although I will admit that a big part of the point of the reals was solutions to polynomials, and it takes complex numbers to be able to solve them all.
Although I will admit that a big part of the point of the reals was solutions to polynomials, and it takes complex numbers to be able to solve them all.
> I find the idea of a universe with a discrete substrate both natural and compelling.
I really like the idea of a finite, ever growing, and totally knowable universe. What Wolfram is searching for. The universe was a small finite graph at the beginning (which we can possibly know by looking at the sky), and it grows since that, according to some deterministic rules. This means that the universe is perfectly replayable, we can replay history, observe the thoughts of famous people, and such.
I really like the idea of a finite, ever growing, and totally knowable universe. What Wolfram is searching for. The universe was a small finite graph at the beginning (which we can possibly know by looking at the sky), and it grows since that, according to some deterministic rules. This means that the universe is perfectly replayable, we can replay history, observe the thoughts of famous people, and such.
I believe there may be ways to introduce entropy into a system like that so it need not completely deterministic, but can’t quite remember the name for the topic…
Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity and honorable mention for Wolfram's hypergraph.
> Maybe someone has a survey paper? Two for discussion would be Loop Quantum Gravity
What would you like to know about it?
> honorable mention for Wolfram's hypergraph.
I'd call it more of a dishonourable mention. If you're looking for niche theories in physics with discrete spacetime, I guess you'd want to look at https://en.wikipedia.org/wiki/Causal_sets
What would you like to know about it?
> honorable mention for Wolfram's hypergraph.
I'd call it more of a dishonourable mention. If you're looking for niche theories in physics with discrete spacetime, I guess you'd want to look at https://en.wikipedia.org/wiki/Causal_sets
Why dishonorable?
His work is pretty interesting to me: graph taxonomy of all possible rules and rules spaces (what Wolfram calls the Ruliad) seems like a very natural extension of Computer science and Analysis.
His work is pretty interesting to me: graph taxonomy of all possible rules and rules spaces (what Wolfram calls the Ruliad) seems like a very natural extension of Computer science and Analysis.
Thanks.
Curious why the hypergraph diss? Regardless of their process, it sounds like they came up with a model which has some alignments with conventional theory.
Curious why the hypergraph diss? Regardless of their process, it sounds like they came up with a model which has some alignments with conventional theory.
Sounds like your describing complex systems thoery https://en.wikipedia.org/wiki/Complex_system
Not quite. I’m describing a system with discrete structure that gives rise to macro level complex system in the same way the game of life can implement arbitrary computation.
The way renormalisation is mostly covered in popular media, made me think you can’t do quantum physics without dirty tricks to get rid of infinites.
Then I read The Road to Reality by Roger Penrose and in the chapter on renormalisation it mentions just “by the way“ that the problem goes away if you assume the universe has a minimum scale, or something like that.
Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation?
Why is the default assumption that the universe is continuous? Doesn’t seem like we have a good reason for it, other than the fact that we use integration as a tool and that assumes continuity by default. Does it make mathematical analysis easier all-in-all? Why not just pick the Planck length as a reasonable assumption for the quantisation of space? Might not be correct but probably a more reasonable guess than assuming that it’s continuous?
Then I read The Road to Reality by Roger Penrose and in the chapter on renormalisation it mentions just “by the way“ that the problem goes away if you assume the universe has a minimum scale, or something like that.
Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation?
Why is the default assumption that the universe is continuous? Doesn’t seem like we have a good reason for it, other than the fact that we use integration as a tool and that assumes continuity by default. Does it make mathematical analysis easier all-in-all? Why not just pick the Planck length as a reasonable assumption for the quantisation of space? Might not be correct but probably a more reasonable guess than assuming that it’s continuous?
> Why is the default assumption that the universe is continuous?
Why is your default assumption that physicists haven't tried to make it discrete? You need to break Lorentz invariance to have the universe be discrete, which is an effort that's worth a Nobel prize.
> Doesn’t seem like we have a good reason for it
We have many good reasons for it.
https://www.forbes.com/sites/startswithabang/2020/04/17/this...
From the article: "If space is discrete, then the principle of relativity is wrong"
Why is your default assumption that physicists haven't tried to make it discrete? You need to break Lorentz invariance to have the universe be discrete, which is an effort that's worth a Nobel prize.
> Doesn’t seem like we have a good reason for it
We have many good reasons for it.
https://www.forbes.com/sites/startswithabang/2020/04/17/this...
From the article: "If space is discrete, then the principle of relativity is wrong"
> other than the fact that we use integration as a tool and that assumes continuity by default.
Is there really no equivalent of calculus that works on a "quantized"/discreet numberline? The seems really hard to believe.
Is there really no equivalent of calculus that works on a "quantized"/discreet numberline? The seems really hard to believe.
> Why isn’t this common knowledge? Why isn’t this one of the first things that gets told when talking about renormalisation?
It's because of the historical development where it really did start off as a nasty bag of mathematical tricks. It wasn't until much later that Wilson and co put renormalization procedures on firm footing, and by that time it had a quite bad reputation.
It's because of the historical development where it really did start off as a nasty bag of mathematical tricks. It wasn't until much later that Wilson and co put renormalization procedures on firm footing, and by that time it had a quite bad reputation.
I thought it was due to relativity which assumes a continuous space. There's a lot of evidence to back up relativity, though it does have issues with not working in a black hole, so it would be questionable to throw it out without something better to replace it.
> For example: how many real numbers are there? The continuum
hypothesis proposes a conservative answer, but since this is independent of the
usual axioms of set theory, the question remains open: there could be vastly
more real numbers than most people think.
If don't agree with that. There are exactly as many real numbers as there are inifnite sequences consisting of zeroes and on (0,0,1,0,1,...). That is the same as the set of subsets of the natural numbers. That is well-defined. The continuum hypothesis asks if there are infinities smaller than this one, but bigger than the infinity of the natural numbers.
If don't agree with that. There are exactly as many real numbers as there are inifnite sequences consisting of zeroes and on (0,0,1,0,1,...). That is the same as the set of subsets of the natural numbers. That is well-defined. The continuum hypothesis asks if there are infinities smaller than this one, but bigger than the infinity of the natural numbers.
No the size of the real numbers is not well defined.
To be well defined means that there is an unambiguous interpretation of a proposition. The fact that the continuum hypothesis is independent of the usual axioms of set theory means that the size of the real numbers is not well defined (within those usual axioms), it's ambiguous whether the size of the real numbers is Aleph_1, Aleph_2, etc....
Just pointing out that the size of the real numbers is equal to the number of possible combinations of an infinite sequence of 0s and 1s doesn't give the size any well defined interpretation, it's just an alternative formulation of the size. Yes there are many ways to formulate the size of the real numbers, but unless you can take one of those formulations and pin it down to a specific and unambiguous value that can be operated on, then the size of the reals remains ambiguous.
It's actually really open ended how large the size of the reals are. All that's known is that it's greater than Aleph_0, and that it's not equal to any limit cardinal such as Aleph_omega.
https://en.wikipedia.org/wiki/Well-defined_expression
To be well defined means that there is an unambiguous interpretation of a proposition. The fact that the continuum hypothesis is independent of the usual axioms of set theory means that the size of the real numbers is not well defined (within those usual axioms), it's ambiguous whether the size of the real numbers is Aleph_1, Aleph_2, etc....
Just pointing out that the size of the real numbers is equal to the number of possible combinations of an infinite sequence of 0s and 1s doesn't give the size any well defined interpretation, it's just an alternative formulation of the size. Yes there are many ways to formulate the size of the real numbers, but unless you can take one of those formulations and pin it down to a specific and unambiguous value that can be operated on, then the size of the reals remains ambiguous.
It's actually really open ended how large the size of the reals are. All that's known is that it's greater than Aleph_0, and that it's not equal to any limit cardinal such as Aleph_omega.
https://en.wikipedia.org/wiki/Well-defined_expression
Do you have a background working with cardinals? I don't, so maybe I'm missing something here.
But why would you consider the reals bigger, if there exists an Aleph_1 which is in between the reals and the naturals? After all, the reals are still the same set?
But why would you consider the reals bigger, if there exists an Aleph_1 which is in between the reals and the naturals? After all, the reals are still the same set?
Can you rephrase your question?
Why would someone consider the reals to be bigger than what, Aleph_1? I'd say the majority of mathematicians consider the reals to be bigger than Aleph_1, including both Kurt Godel (believed it to be equal to Aleph_2) and Paul Cohen.
Paul Cohen gave a talk where he said he believes the continuum is something unimaginably large, in fact if we take C to be the size of the continuum, then Cohen believes C = Aleph_C.
There are a variety of motivations for these reasons, but they mostly boil down to believing that the continuum is something that is unreachable and non-constructible.
Why would someone consider the reals to be bigger than what, Aleph_1? I'd say the majority of mathematicians consider the reals to be bigger than Aleph_1, including both Kurt Godel (believed it to be equal to Aleph_2) and Paul Cohen.
Paul Cohen gave a talk where he said he believes the continuum is something unimaginably large, in fact if we take C to be the size of the continuum, then Cohen believes C = Aleph_C.
There are a variety of motivations for these reasons, but they mostly boil down to believing that the continuum is something that is unreachable and non-constructible.
A set M is said to have the same size as the reals R if there is a bijection M -> R. M has size less or equal than R (|M| <= |R|) if there is an injection M -> R, and |R|<=|M| if there is an injection R -> M. That is my working definition of "size".
The continuum hypothesis asks if |R| = Aleph_1, or not? If not, then Aleph_1 < |R|, meaning there is an infinite set M with an injection M -> R, but no bijection to R or the natural numbers N (|N|=Aleph_0).
But the existence or non-existence of such a set M does not at all change the size of R in my mind. Why would it?
The continuum hypothesis asks if |R| = Aleph_1, or not? If not, then Aleph_1 < |R|, meaning there is an infinite set M with an injection M -> R, but no bijection to R or the natural numbers N (|N|=Aleph_0).
But the existence or non-existence of such a set M does not at all change the size of R in my mind. Why would it?
I suppose it's not too clear to me what you mean then, but of course this is an abstract topic so that's to be understood.
A size just by itself is meaningless, if I said I have Aleph_0 units of happiness well for all you know I could be depressed, even if I said I have 2^Aleph_0 units of happiness I could still be depressed. It's only if you can compare what those units of happiness are to things you are familiar with, like how many units of happiness is eating chocolate, or being able to rest after a hard and productive day at work, that you would be able to get a sense of how happy (or depressed) I am.
So sure, just knowing the size of the real numbers on its own without comparing it to anything doesn't change its significance in and of itself because it's meaningless. But if we could begin to identify certain sets that were smaller than the size of the reals (and larger than the naturals), and we could identify properties of these sets that gave us a sense of their expressive power, their reasoning power, etc... then we might start to appreciate just how large the real numbers are.
We know categorically that Aleph_1 is the size of the set of the first uncountable ordinal number w_1, so any ordinal less than w_1 is countable. If the size of the continuum is Aleph_1 then that would have rather unintuitive implications involving the types of functions that are possible to define and reason about along with their relationship to probabilities and randomly choosing real numbers within the interval [0, 1] depending on whether the size of the real numbers is Aleph_1 or bigger. Probably one of the more accessible arguments for why this is is linked below [1].
[1] https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
A size just by itself is meaningless, if I said I have Aleph_0 units of happiness well for all you know I could be depressed, even if I said I have 2^Aleph_0 units of happiness I could still be depressed. It's only if you can compare what those units of happiness are to things you are familiar with, like how many units of happiness is eating chocolate, or being able to rest after a hard and productive day at work, that you would be able to get a sense of how happy (or depressed) I am.
So sure, just knowing the size of the real numbers on its own without comparing it to anything doesn't change its significance in and of itself because it's meaningless. But if we could begin to identify certain sets that were smaller than the size of the reals (and larger than the naturals), and we could identify properties of these sets that gave us a sense of their expressive power, their reasoning power, etc... then we might start to appreciate just how large the real numbers are.
We know categorically that Aleph_1 is the size of the set of the first uncountable ordinal number w_1, so any ordinal less than w_1 is countable. If the size of the continuum is Aleph_1 then that would have rather unintuitive implications involving the types of functions that are possible to define and reason about along with their relationship to probabilities and randomly choosing real numbers within the interval [0, 1] depending on whether the size of the real numbers is Aleph_1 or bigger. Probably one of the more accessible arguments for why this is is linked below [1].
[1] https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
The constructable numbers, which is what you are describing are aleph naught, or countable.
The Cantor set is a way to think about this.
The Cantor set is a way to think about this.
Not true. If you think about binary representation of numbers it is clear that there are as many infinite binary sequences as there are real numbers between 0 and 1. So that is uncountably infinite. Alternatively, apply the usual diagonal argument directly to the set of sequences to see that there are uncountably many of them.
This has not much to do with constructible numbers, which are those (complex) numbers which can be constructed using compass and straightedge constructions starting from 0 and 1.
This has not much to do with constructible numbers, which are those (complex) numbers which can be constructed using compass and straightedge constructions starting from 0 and 1.
The definable numbers, no matter if you use the constructible, algebraic, or computable numbers are countable infinities, or aleph naught.
Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity.
I realize that those concepts are difficult to build intuitions around, but what I said was correct.
definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers.
You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals.
The reals being uncomputable 'almost everywhere' is well proven at this point.
Cantor diagonalization being the typical way of showing it.
'Almost everywhere' as used above has a formal definition:
A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero.
The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...
Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity.
I realize that those concepts are difficult to build intuitions around, but what I said was correct.
definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers.
You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals.
The reals being uncomputable 'almost everywhere' is well proven at this point.
Cantor diagonalization being the typical way of showing it.
'Almost everywhere' as used above has a formal definition:
A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero.
The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...
Sure. What I took offense at was the "which is what you are describing" part. I described uncountable infinities.
Sorry, the way it was described I didn't take as the power set but the constructable sets.
Previous discussion (2016) - https://news.ycombinator.com/item?id=12441990
So, I only escaped with a bachelor's in physics and a bit more math than was usually required for the degree, but I have often wondered if "we" went astray by spending time on concepts like infinity. Summing 1 + 2 + 3 + 4 + ... to get -1/12 using various infinities and "regularization" just seems, well, like some kind of sleight of hand, and certainly divorced from physical reality.
Ramanujan summation (1 + 2 + ... = -1/12) is used in the derivation of Casimir effect I'm afraid.
I wonder if there's an alternate way to get there without that monstrosity, which manages to be triply-offensive.
As far as I know, just like matter and energy, information density can also contribute to a black hole.
A real number has infinite precision, so even a single particle with a real number position or weight, would have infinite information and create a black hole.
Would that be a valid reasoning to say why it can't be real numbers?
A real number has infinite precision, so even a single particle with a real number position or weight, would have infinite information and create a black hole.
Would that be a valid reasoning to say why it can't be real numbers?
You got a glib answer from another guy, but I like the question. It's kind of like you're saying: how much information can I store in a particle? And should there be a limit?
(I'm assuming you're talking about in the classical case, since the article also also considers classical dynamics and the problems associated with it.)
I'm actually gonna make a note to think about this later because this reminds me a lot of Charles Bennett's work on information and entropy, and I don't think it's a bad question at all. I'm considering the idea of representing the particle's position as x = .01010111001 and then idealizing a "machine" that lets it do work by erasing one bit of information. I have to work so I can't think about this right now but it's a cool thought.
Edit: I did think about and you're right! You can extract infinite energy if you know a particle's position to perfect precision. Very interesting!
Make the particle an ideal gas in a box at temperature T. At some instant in time, measure the position of the particle to N bits precision. Now place membranes around the particle to keep it to within the precision we measured it. We can kill one bit of precision by allowing the particle to isothermally expand the membranes into a volume with half the precision that we measure (so if we measure down to 1mm precision, we let it expand into 2mm.) The amount of work we can extract is:
W = \int p dV = \int kT dV V = kT ln 2
So for N bits precision you can extract NkT ln2 units of energy. I'll leave it to someone else to comment on whether this makes a black hole or not but this is exactly in the spirit of the article!
(I'm assuming you're talking about in the classical case, since the article also also considers classical dynamics and the problems associated with it.)
I'm actually gonna make a note to think about this later because this reminds me a lot of Charles Bennett's work on information and entropy, and I don't think it's a bad question at all. I'm considering the idea of representing the particle's position as x = .01010111001 and then idealizing a "machine" that lets it do work by erasing one bit of information. I have to work so I can't think about this right now but it's a cool thought.
Edit: I did think about and you're right! You can extract infinite energy if you know a particle's position to perfect precision. Very interesting!
Make the particle an ideal gas in a box at temperature T. At some instant in time, measure the position of the particle to N bits precision. Now place membranes around the particle to keep it to within the precision we measured it. We can kill one bit of precision by allowing the particle to isothermally expand the membranes into a volume with half the precision that we measure (so if we measure down to 1mm precision, we let it expand into 2mm.) The amount of work we can extract is:
W = \int p dV = \int kT dV V = kT ln 2
So for N bits precision you can extract NkT ln2 units of energy. I'll leave it to someone else to comment on whether this makes a black hole or not but this is exactly in the spirit of the article!
Alright: the kicker (and the reason why I don't think this will result in a black hole) is that in order to extract that energy from the particle, you have to continuously re-heat it and add more energy back each time you want kT \ln 2 units of work done. Remember that the work is done isothermally, so the particle's energy is actually constant throughout this process.
What's really happening is that if we consider the positional information to be continuous, then the particle's positional entropy is basically infinite. We can reduce this entropy (locally, to the particle) with a measurement, and use that lowered entropy to extract as much isothermal work as we want. But its energy density is unaffected by the positional information.
What's really happening is that if we consider the positional information to be continuous, then the particle's positional entropy is basically infinite. We can reduce this entropy (locally, to the particle) with a measurement, and use that lowered entropy to extract as much isothermal work as we want. But its energy density is unaffected by the positional information.
The amount of information in a real number is not determined by the number of digits. For example, 1/3 = 0.333333… but we wouldn’t say 1/3 has “infinite information.”
Perhaps a better way to characterize the amount of information in a number is to look at the length of the program necessary to compute it. Of course, this depends on the programming language, but I think it’s fair to say that any particular program to compute a number is in some loose sense an upper bound on the information in the number.
So, most of the irrational numbers we’re familiar with (pi, e, phi) are rather low on information since they may be computed by very short programs, despite having infinitely many non-repeating digits.
Perhaps a better way to characterize the amount of information in a number is to look at the length of the program necessary to compute it. Of course, this depends on the programming language, but I think it’s fair to say that any particular program to compute a number is in some loose sense an upper bound on the information in the number.
So, most of the irrational numbers we’re familiar with (pi, e, phi) are rather low on information since they may be computed by very short programs, despite having infinitely many non-repeating digits.
Yes, but the real numbers that can be computed to arbitrary precision with an algorithm are countable: this means almost all the reals contain an infinite amount of information. The real numbers are useful to reason and construct proofs, but as far as the physical world go, only very few of them are actually meaningful.
The non-computable reals are also not uniquely representable. It’s this property which I think is the most reasonable explanation for why these real numbers don’t cause an information black hole.
But 1/3 counts as discrete in a sense, you can describe it with a finite amount of symbols.
To move from one point to another, a particle has to move through all of them though, including the non computable ones
To move from one point to another, a particle has to move through all of them though, including the non computable ones
If you're talking about a particle in real life, then you're talking about quantum mechanics, which is discrete, so the real numbers don't even come into play in that realm. Then, the use of real numbers in physics is a conceptual and calculational convenience which is not quite correct.
Particles don't have infinite precision. Position and momentum precision is limited by Heisenberg uncertainty principle. Weight precision is also limited because energy is quantized.
That, however, doesn't mean that spacetime is not a continuum.
That, however, doesn't mean that spacetime is not a continuum.
Then what does spacetime being a continuum mean?
What's the point of being a continuum if nothing in it can have arbitrary values. If everything in the spacetime has discrete values, what's there still to be continuous?
What's the point of being a continuum if nothing in it can have arbitrary values. If everything in the spacetime has discrete values, what's there still to be continuous?
Gravity is continuous, and to disprove that you would need to disprove Lorentz invariance and Einstein's relativity.
It's actually amusing that, just few weeks ago, people here on HN were arguing on why are we trying to quantize gravity and that it makes more sense for it to be non-quantum.
It's actually amusing that, just few weeks ago, people here on HN were arguing on why are we trying to quantize gravity and that it makes more sense for it to be non-quantum.
But gravity is caused by the discrete particles (those with mass), so is it really continuous then? A discrete particle could only move in discrete steps, causing discrete jumps in gravity, would that be right?
And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
> But gravity is caused by the discrete particles (those with mass), so is it really continuous then?
Of course the field can be continuous, even if the weight is exactly some rational number. "Discrete steps" just means that the value differences in gravitational waves is also discrete. A bigger problem, where relativity clashes with QM, is stuff like singularity and superposition. We don't have an answer for that yet.
Of course the field can be continuous, even if the weight is exactly some rational number. "Discrete steps" just means that the value differences in gravitational waves is also discrete. A bigger problem, where relativity clashes with QM, is stuff like singularity and superposition. We don't have an answer for that yet.
It is, this idea is explored a bit in this paper: https://arxiv.org/abs/1803.06824. It also argues that using real numbers for physical quantities makes classical mechanics misleadingly deterministic, that is, they're hiding information that is impossible to measure in the initial conditions.
I'm not certain, but I think the only fundamental thing contributing to a black size is energy density[0], mass only contributes because it has an energy density (that's what E=mc^2 is), and information because it has to be carried by stuff which itself has a minimum energy per bit.
If so, no: if any particle were to need infinite precision to be described, that would just mean we were wrong about maximum information density within a minimum boundary surface.
[0] while both charge and angular momentum reduce it
If so, no: if any particle were to need infinite precision to be described, that would just mean we were wrong about maximum information density within a minimum boundary surface.
[0] while both charge and angular momentum reduce it
Particle positions aren't stored by the universe as REAL but rather as DECIMAL(62, 35)
Infinite precision does not imply an infinite amount of information.
No.
> No.
Please elaborate
Please elaborate
Because what you're describing is coordinate or unit system dependent. I could invent a unit system where the mass of an electron is pi units, and another unit system where the mass of the electron is 1 unit. Me using a different unit system doesn't affect the physics of the system.
But the particles can move around, they won't always be exactly on values of pi, 1, etc... in any coordinate system. And pi and 1 happen to be examples that can be described with a finite amount of symbols. But a continuous particle has to move through any value in between.
The point is that you're discussing precision as it relates to a coordinate system trying to model position.
That really and truly has no bearing on reality except insofar as the coordinate system has attempted to map reality.
That really and truly has no bearing on reality except insofar as the coordinate system has attempted to map reality.
Your idea interests me.
I think it's related to the Plank Scale, but what you're talking about would possibly be a "Plank Precision"?
I'm having a hard time seeing if this is fundamentally the same thing as the Plank Length, or if it's something different entirely.
https://en.wikipedia.org/wiki/Planck_units#Planck_length
Very cool idea though.
I think it's related to the Plank Scale, but what you're talking about would possibly be a "Plank Precision"?
I'm having a hard time seeing if this is fundamentally the same thing as the Plank Length, or if it's something different entirely.
https://en.wikipedia.org/wiki/Planck_units#Planck_length
Very cool idea though.
If you look at the history of physics, including not just the final breakthroughs but the previous misunderstandings, there's a long history of "Well, we thought it had to be either A or B, but it turns out it's a novel and hard-to-imagine combination of both of those things."
For example, consider the universal speed limit that we now call the speed of light. Prior to relativity, when space and time were kept rigidly separated, there was a bit of a problem. The math of Newtonian physics and their corresponding frame transforms admit no particular speed limit; no matter how fast a thing is going, a force could always be applied to it to make it go yet faster. So a speed limit seemed to be impossible. But observationally, it became more and more clear there was one, and what's more, it seemed to be unrelated to your frame of reference. So apparently there was a speed limit, and not only that, but one that was essentially uncharacterizable in Newtonian physics. But if something was going the speed limit, what happened if you applied further forces on that thing?
The universe seemed contradictory; it seemed to have a speed limit, yet it couldn't have one, and if it did have one it seemed it would lead to contradictions as well. So, does the universe have a speed limit or not?
And the answer is basically that that question, in the context of a Newtonian view of the universe, is simply ill formed and has no meaning, because the universe isn't Newtonian. With the better (if not necessarily entirely correct) view that Einstein introduced, what we get is actually something that even now a lot of people don't really get, which is that the universe both does and does not have a speed limit. The former is generally understood; c is the limit. You can not witness anything in the universe going faster than c from your perspective. But there is also a sense in which the universe doesn't have a speed limit; no matter how fast you are going, you can always accelerate in any direction and go faster in that direction. There isn't a speed limit in the sense that there is no speed that you can be going where the universe will say "Nope, stop, you can't go any faster that way because physics won't allow it." (Other things, like blueshifting the cosmic microwave background into arbitrarily high-energy gamma rays from your point of view may stop you, but "physics" itself won't.) Subjectively, time dilation means that there is a sense in which you can head to distant location as fast as you like; you can cover what was originally a million light years in arbitrarily short amounts of subjective time, and there is a real sense in which that becomes your "speed" and it can greatly exceed lightspeed in your original reference frame. And just try explaining to a 17th century physicist the contortions the universe will go through in order to make it that even that arbitrarily-fast object from a certain point of view will still not witness anything going faster than light, with length contraction, time dilation, all that jazz. If you hadn't been taught it, you wouldn't ever come with it on your own.
That is, it's a complicated mixture of both having and not having a speed limit that, when fully understood, makes sense. But if you try to put the answer in the context of the original Newtonian understanding it makes no sense.
Personally I expect the base of our universe to have something similar when it comes to "is it continuous or discrete?" I expect it's going to end up being something that has elements of both, in some manner that is essentially impossible to understand if you insist on fitting it into that strict dichotomy but all works together and makes sense on its own terms. Loop quantum gravity is the best effort in this direction I've seen, but since it isn't "done" yet doesn't seem to be the right solution.
(There's been a lot of things in science history where there was some sort of "Is this thing rigidly This or That?" and the answer turns out to be "well, yes, and also no, it's more complicated and fluid than you thought" in some manner that would have been difficult to guess in advance.)
For example, consider the universal speed limit that we now call the speed of light. Prior to relativity, when space and time were kept rigidly separated, there was a bit of a problem. The math of Newtonian physics and their corresponding frame transforms admit no particular speed limit; no matter how fast a thing is going, a force could always be applied to it to make it go yet faster. So a speed limit seemed to be impossible. But observationally, it became more and more clear there was one, and what's more, it seemed to be unrelated to your frame of reference. So apparently there was a speed limit, and not only that, but one that was essentially uncharacterizable in Newtonian physics. But if something was going the speed limit, what happened if you applied further forces on that thing?
The universe seemed contradictory; it seemed to have a speed limit, yet it couldn't have one, and if it did have one it seemed it would lead to contradictions as well. So, does the universe have a speed limit or not?
And the answer is basically that that question, in the context of a Newtonian view of the universe, is simply ill formed and has no meaning, because the universe isn't Newtonian. With the better (if not necessarily entirely correct) view that Einstein introduced, what we get is actually something that even now a lot of people don't really get, which is that the universe both does and does not have a speed limit. The former is generally understood; c is the limit. You can not witness anything in the universe going faster than c from your perspective. But there is also a sense in which the universe doesn't have a speed limit; no matter how fast you are going, you can always accelerate in any direction and go faster in that direction. There isn't a speed limit in the sense that there is no speed that you can be going where the universe will say "Nope, stop, you can't go any faster that way because physics won't allow it." (Other things, like blueshifting the cosmic microwave background into arbitrarily high-energy gamma rays from your point of view may stop you, but "physics" itself won't.) Subjectively, time dilation means that there is a sense in which you can head to distant location as fast as you like; you can cover what was originally a million light years in arbitrarily short amounts of subjective time, and there is a real sense in which that becomes your "speed" and it can greatly exceed lightspeed in your original reference frame. And just try explaining to a 17th century physicist the contortions the universe will go through in order to make it that even that arbitrarily-fast object from a certain point of view will still not witness anything going faster than light, with length contraction, time dilation, all that jazz. If you hadn't been taught it, you wouldn't ever come with it on your own.
That is, it's a complicated mixture of both having and not having a speed limit that, when fully understood, makes sense. But if you try to put the answer in the context of the original Newtonian understanding it makes no sense.
Personally I expect the base of our universe to have something similar when it comes to "is it continuous or discrete?" I expect it's going to end up being something that has elements of both, in some manner that is essentially impossible to understand if you insist on fitting it into that strict dichotomy but all works together and makes sense on its own terms. Loop quantum gravity is the best effort in this direction I've seen, but since it isn't "done" yet doesn't seem to be the right solution.
(There's been a lot of things in science history where there was some sort of "Is this thing rigidly This or That?" and the answer turns out to be "well, yes, and also no, it's more complicated and fluid than you thought" in some manner that would have been difficult to guess in advance.)
It seems that most professional mathematicians have never really given it much thought (or at least no more than other ontological, ‘non-mathematical’ questions). Of course, when it comes to fundamental physics it all starts to become more relevant.
edit: My comment shouldn’t be taken as a recommendation. I’m not suggesting mathematicians should worry about such things (it may well be a waste of their time!). I’m simply surprised that more don’t find it as unnerving as I did at the time.