Biocurious is a weblog about biology, quantified.

Science is like the real number line

by PhilipJ on 19 September 2006

Shamelessly quoting RRResearch, the PI from a lab I’m collaborating on a project with:

Scientific knowledge is like the “real number line” used in introductory math classes. In the line, every point is a number, but no matter how close together two points (or numbers) are, there are always infinitely many other points separating them. The real world similarly contains infinitely many things to discover. No matter how much we find out about something, there are always many more important things to find out. I use this analogy for beginning science students, who are often concerned that all the important discoveries have already been made.

This is a totally awesome way to think about science. Every question answered leads to new questions to tackle. Whether you agree with John Horgan’s hypothesis in The End of Science that all the fundamental discoveries in science are complete, I think it would be hard to argue that all the important discoveries in science have been made. There are always going to be things we don’t understand, and there will always be enquiring minds trying to figure things out.

  1. Nate    4353 days ago    #

    The metaphor seems a little off.

    Even though the physical universe may be finite (and discrete in all 4, or 11, or whatever dimensions? help me out here), I’ll grant that one could argue, perhaps convincingly, that there are an infinite set of facts to be learned.

    But how does the purported denseness (the property of the reals that the author of the RRResearch post is fascinated with) of that knowledge come in? Why is the set of possible scientific facts like the reals and not like some non-dense, but infinite structure like the integers? Or could the set of (semantic) facts be finite? How do we interpret the ordering of facts suggested by this metaphor? Is an ordering, which implies a kind of progression, evidence of teleology? Design? Purpose?

    Non-scientists often use metaphors from science poorly—e.g., Heisenberg’s uncertainty principle is a favourite; let’s not make the same mistake!

  2. Alejandro Rivero    4353 days ago    #

    actually I am not impressed about someone speaking of science and naming the real line as “points (or numbers)”, when a point in the real line is not a number but a [class of equivalence of] pair of series of number converging one to another :-D

  3. Bill Tozier    4353 days ago    #

    You try to be inspiring, and all you get is critics: There’s a typo in the outgoing link to Rrresearch….

  4. JoeK    4353 days ago    #

    Alejandro, actually you can define real numbers perfectly well axiomatically as a complete Archimedean ordered field without having to worry about their construction in terms of equivalence classes. Although the construction of the reals from the integers is kind of cool, the pre-occupation with it is left over from the nineteenth century when there was some anxiety about whether or not real numbers “really” existed.

    As for the analogy, I’m not convinced it’s that great for begining students. If you’ve studied some math then you can be seriously impressed by the reals – the interleaving of the rationals with the irrationals, all set against the “dark matter” of uncomputable numbers is really quite bizarre. (In that sense it’s certainly better than an analogy with the rationals, for which the argument would go through as well.)

    However, I worry that a beginning student could take the analogy to mean that new questions become ever narrower and smaller. This is quite literally a picture of science as refining measurements to the nth decimal place, a caricature of exactly the line John Horgan argues!

    You could try spicing it up with an example like the Mandelbrot set, which is quite intriguing in its capacity to throw up “variations on a theme” as you zoom in.

    Altogether I think the important thing to convey is the sense that “the more we know, the less we know”. Some people use this as an antiscience point, but I would stress that we really do know much, much more. A good explanation forces the student to try to hold the two contradictory dynamics in mind at once. Properly understanding science will blow your mind.

  5. PhilipJ    4353 days ago    #

    Hi Bill – thanks, properly linked now. Maybe I shouldn’t even try in the future.

    JoeK – in some ways, I’m agreeing with John Horgan’s hypothesis but disagreeing with his conclusion. There really are only so many truly fundamental discoveries to make, but I don’t think that means science is about to end, precisely because fundamental and interesting are not the same thing.

    New questions do get ever more specific, but this doesn’t change their importance for understanding lower level things as well. As per all analogies, the real number line as science doesn’t quite do it justice (and thinking about science as fractal in nature is something I’ve thought about before too, and may be a little more appropriate), but I still think it captures the basic idea fairly well. Those not working on fundamental theories are working on specific problems, but there is nothing wrong with specific problems that are interesting, and as you’ve said, even with specific problems, the more we know the less we know.

  6. Uncle Al    4352 days ago    #

    If you omit the imaginaries you lose a bunch of roots, alternating current, Special Relativity, and Euler’s link between algebra and analytic geometry.

    Think outside the torioid. (Don’t get all excited! Its triangles’ interior angles sum to 180 degrees.)

  7. Doug    4352 days ago    #

    I agree with Uncle Al.

    However, there is probably a misnomer in using the term “imaginaries”.

    Too many people interpret “imaginariesâ€? as equivalent to a ‘mathematical construction’ of not real numbers.

    Euler’s link [identity circle] demonstrates that these numbers exist and probably should be referred to as either invisible or intersecting numbers [conserving ‘i’] or perhaps even as juxtaposed numbers [conserving ‘j’].

    If the square root of +1 were treated as the square root of -1, then we might have:
    u = -1 union +1, for square root of +1
    i = -1 intersection +1, for square root of -1.

    In Euler’s link, contrasted or juxtaposed numbers are diagrammed.

    Yet such numbers may not be visible to humans due to some reason such as wave length or polarity.

    There may even be a situation similar to that of “invisible” tree rings in tropical trees as discussed in ‘Bright sparks reveal invisible tree rings:
    Calcium markers could aid climate studies in the tropics’.

  8. JoeK    4352 days ago    #


    It’s just the gut reaction I had to the real number line analogy, so it’s completely subjective. I think it’s too likely to conjure up the idea that the new questions left to science are sequencing the 346272th base pair in yet another genome or solving yet another crystal structure.

    But your idea that today’s science is characterised by questions becoming more specific seems to me the exact opposite of the “conventional wisdom”. The whole sense in the air (that you can see e.g. in the Theory of Biology thread below) is that biology has reached a moment at which we can generalise. All the good work of previous decades has generated mountains of specific data that we are now in a position to theorise.

    Look at the RRResearch post you linked to. It’s about the mechanism by which purine nucleotides inhibit both transformation and the induction of two competence genes. But is it really? The further questions it raises, among others, are about secondary structure of RNA, microRNA, and riboswitches. This is quickly headed into a dense web of results – RNA has been at the heart of molecular biology in the last decade.

    And when we put together all the many thousands of incremental results we may discover that we are on the path to convincing evidence for an RNA world. Of course, we’re not there yet. But I think the origin of life was a specific example that Horgan used in his book as a question that’s forever beyond reach. I don’t think that’s true, and I think it’s important to show beginning students how mastering the details can still help with the “big questions”.

    Maybe you don’t think that the origin of life is a “fundamental” question? I’m not really sure is meant by “fundamental” anyway. If we go all the way with the Anderson / Gell Mann / Laughlin “more is different” programme, then fundamental isn’t defined through reductionism. Even if I don’t go all the way with them, I think they have some strong points. (Horgan has rightly been critical of the hype surrounding what he calls “chaoplexity”. But as I recall he hasn’t really engaged with the idea that the renormalisation group provides a paradigm for this sort of thinking, which is a stronger basis for progress.)

    In fact even before complexity, Feynman had already grasped some limits of “fundamental” physics with his usual good sense:,,1486637,00.html:

    Anyway, enough of this rambling. The gamble I’m taking is that general theories (built, of course, from specific problems) are about to make biology very interesting – I’m days away from embarking on a PhD after years away from university, moving from theoretical physics to experimental chemistry…

    Uncle Al,

    OK, I’ll bite. Why do you worry about losing complex roots, but not quaternions or any other field extension? For special relativity do geometry in Minkowski space – imaginary time is only useful for obscure applications like continuation of Euclidean path integrals. AC circuits are a superficial use of complex numbers. For the real thing, see Penrose: do the link to relativity properly (the Lorentz group is secretly mobius transformations of the complex plane) and, above all, understand that complex numbers are the heart of Quantum Mechanics. Understand that the magic of complex numbers is in analysis. Euler’s formula is brilliant, but Cauchy’s theorem is deeper and Picard’s theorem is more startling. Once you have mastered complex analysis you can have not just tori (elliptic functions) but any Reimann surface with whatever curvature you like. (And of course I already had complex numbers covered with the Mandelbrot set.)

  9. Uncle Al    4351 days ago    #

    Published acreage is expensive. Make your point, support it, move on. The list of good stuff off the real number line is very large. Uncle Al is fond of non-commutative relationships (chirality). A two-day experiment could falsify all of physics on a technicality for less than $100 of consummables,

    Do you have two calorimeters hard by 45 degrees latitude for two days? Nothing requires space to be isotropic to chirality. Nobody has ever looked at an extreme divergent case.

    You won’t see something different unless you look elsewhere.

  10. Rosie Redfield    4351 days ago    #

    Back off, guys! This is meant to be a SIMPLE analogy for FIRST YEAR science students, and I’m a molecular biologist, not a mathematician.

    Analogies only work when they’re to something simple that the listener already understands. Even if I understood Mandelbrot sets, nothing would be gained by including them in my analogy because the first-year students wouldn’t understand them well enough to get the analogy.

    And note that I said “important things to find out” (i.e. not just another base pair).

  11. JoeK    4351 days ago    #


    You’re right that simple analogies are important in teaching. If that one clicks with you then I’m sure you’ll teach it well, with an emphasis on important things.

    Maybe because I started as a physicist that my own first response was to remember Lord Kelvin’s remark in 1900 that the only task left for physics was more precise measurement. Kelvin was the most famous then living physicist, but missed that relativity and quantum were about to change the story. I think that makes me allergic to anything that could be taken as saying truth is in the 12th decimal place – and so I took the analogy a bit too literally. If I had to teach it I suppose I would have to stop the over-analysis and step back to reflect for a few moments. Hopefully then I could use it too.

    Sorry for the blather about complex numbers. Maybe it’s something about blog comment threads that they can tend to meander.

  12. JoeK    4351 days ago    #

    Also, RRResearch is a really great blog. I haven’t seen anyone else able to blog day to day research like that.

  13. Rosie Redfield    4350 days ago    #

    I suspect lots of scientists could write fascinating blogs about their research if they weren’t inhibited by fear that competitors would steal their ideas or scoop their papers. Science would be both more visible and more productive if they took the risk.

  14. Apocalypse The Dark    3661 days ago    #

    I cant help but find fault in your logic.
    Science should be used for one thing only, to solve relavent modern problems that are imperative to the survival of the human race. Childish ideas and irrelavent thoeries have no place in the modern world which the writer of the article seems to have forgot.Fancy comparisons of science and the real number system will not solve relavent problems like AIDS,Cancer,Pollution, or a fuel to replace gasoline.
    Not to offend anyone but what it comes down to is that your ideas seem to be thought up by intelligent but slopy, unfocused, unorganized children.
    If all of you could apply your knowledge,time,money, and efforts logically instead putting all your efforts into meaningless comparisons, just think what could be accomplished.
    I put this question to all of you!
    How does the number line apply to ANY modern problems that demand an answer?

  15. Andre    3661 days ago    #

    Apocalypse, even scientists need hobbies…

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