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I remember telling my tutor in the first year of a (pure) mathematics degree that I didn’t really enjoy analysis because I “couldn’t see why anyone should belie

• agnosticmantis 3y ago
  Bring us another complete ordered field and I’ll pay handsomely for it.
• whatisyour 3y ago
  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.
  • xanderlewis 3y ago
    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.
    • red_trumpet 3y ago
      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.
      • xanderlewis 3y ago
        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.
    • whatisyour 3y ago
      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.
    • samus 3y ago
      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.
  • ben_w 3y ago
    > 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).
  • nyrikki 3y ago
    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.
• eigenspace 3y ago
  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.
  • xanderlewis 3y ago
    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.
    • 4death4 3y ago
      > 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.
      • xanderlewis 3y ago
        > 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.
        • pdonis 3y ago
• samus 3y ago
  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.
• bmacho 3y ago
  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.
  • BoiledCabbage 3y ago
    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).
• 613style 3y ago
  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.
  • tutfbhuf 3y ago
    "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.
    • samus 3y ago
      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!
      • tutfbhuf 3y ago
        "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.
    • 613style 3y ago
      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.
• red_trumpet 3y ago
  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.
  • danbruc 3y ago
    You probably mean f(1) > 0.
• oslac 3y ago
  Any number that cannot be effectively computed does not exist; there are actual hard physical limits to Numbers.
  • r2_pilot 3y ago
    I disagree with your statement and provide Graham's number as a counterexample. https://en.m.wikipedia.org/wiki/Graham%27s_number https://en.m.wikipedia.org/wiki/Graham%27s_number (edit: and TREE(3) )
• alberto_ol 3y ago
  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
• vprcic 3y ago
  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?
• ajkjk 3y ago
  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.
• ogogmad 3y ago
  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.
  • zozbot234 3y ago
    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/ 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.
• hbrav 3y ago
  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.
  • rstuart4133 3y ago
    > 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.
• KolenCh 3y ago
  > 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.
  • xanderlewis 3y ago
    > 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.