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).
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.
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.
Of course, this is opening another whole can of worms that is perhaps starting to become off-topic.
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.
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.
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.
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.
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.
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.
Whether or not they're a good description of coordinates in spacetime is a pretty separate matter.
You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.
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.
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.
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.
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, 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.
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.
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".
Can you give an example?
If you can't, doesn't that in itself answer your question?
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?
One interesting fact is that there is a school of mathematics called constructive mathematics. This branch of mathematics proves things without using the law of excluded middle.
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.
I mean I could just say let us deny ∀x(x=x) and run with it, see what happens. It really depends on how one understands the word imagine in this context, I could postulate this but I can not really make sense of a thing not being equal to itself.
And if we can't even consistently think about such a thing, what's the point?
I agree with you that mathematics is pretty much divorced from reality, that mathematics makes up axioms and inspects to consequences of them, and also that some mathematical structures are useful to model aspects of the universe. Some were even specifically invented to model certain aspects of the universe, they were inspired by reality.
But at the bottom of mathematics there is somewhere the logic that underpins everything and one can ask the same questions about its relation with reality. Did we arbitrarily pick the axioms of logic or did we pick them because they make sense in our universe? Or are they universal and there can not be any universe that defies our logic? The question is essentially the following. While we - or at least some people - think that mathematics is quite independent of the real world in general, might this be less true than we think because we accidentally introduced aspects of our universe into it by building on top of a logic that reflects how our universe works? Or the other way around, are there other versions of mathematics build on top of other versions of logic that seem illogical to us but only because our universe instilled us with a certain logic.
Of course they do. That is perfectly consistent mathematically with there being two time dimensions. Two time dimensions is a property of the spacetime geometry, not of individual worldlines.
> whether you would want particles to trace out world surfaces
That would be a different mathematical structure, where "particles" would have something like "two time evolutions" instead of one. What that would imply about the spacetime geometry and its dimensions is a different question.
> I could just say let us deny ∀x(x=x) and run with it
How would you even do that? How would you even make a consistent mathematical structure with that as an axiom? And if you can't, doesn't that mean you can't deny that proposition?
> I could postulate this but I can not really make sense of a thing not being equal to itself.
You're contradicting yourself. If you can't make sense of it, you can't postulate it. "Postulate" doesn't mean "throw some words together". It means "take as an axiom and develop a consistent mathematical structure based on it". If you can't make sense of it, you can't develop a consistent mathematical structure based on it.
> Did we arbitrarily pick the axioms of logic or did we pick them because they make sense in our universe?
If we can't develop a consistent mathematical structure based on any other set of axioms, what choice do we have?
Are you sure that it is impossible to build something consistent on top of denying the law of identity?
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.
[1] https://plato.stanford.edu/entries/philosophy-mathematics/
This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...
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.
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!
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.
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.
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.
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?).
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.
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.
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.
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.
(Though it is very interesting!)
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!
I do agree, but that we don't do that in reality is exactly my point.
R. Hersh, 1979
https://en.m.wikipedia.org/wiki/Graham%27s_number (edit: and TREE(3) )
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.
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 (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.