17 Equations that Changed the World

Yesterday I posted about a map that “changed the world”. Clearly the world changed a lot and for many different reasons because when I got home I noticed the following picture on Facebook, depicting 17 equations that also “changed the world”:


17 Equations

This is from a book by mathematician Ian Stewart.

Of course it’s actually 20 equations, because there are four Maxwell Equations. It is an interesting selection. Are there any surprising omissions?




25 Responses to “17 Equations that Changed the World”

  1. Indeed, there ARE! Why is he *not* listing Einstein’s relativistic field equations of gravitation from Thu, 25-11-1915, which recently celebrated their 100th birthday? After all, most physicists think that these have something to do with (the evolution of) the entire Universe! (He does not hesitate to list Newton twice, does he?) And what kind of Maxwell’s equations of electromagnetism are these supposed to be? (That’s neither the general case, nor the special vacuum case he’s referring to here.) Great idea, but could have been more carefully executed.

  2. Anton Garrett Says:

    I’d add the diffusion equation, conservation of energy, F=ma, the Lorentz transformation, the GR field equations and the Dirac equation.

    • How about the Bayes Theorem?

      • Anton Garrett Says:

        OK but the sum and product rules combined (as they can be). And how about de Morgan’s theorem in binary logic (due actually to William of Ockham…) Let’s also add Euler’s equation in the form exp (i\pi)=-1.

        Bob May wrote a fine short review of chaos theory in Nature in 1975. I’m not sure if he did the main work on the logistic equation (given in the list next to his name) but in any case the key point, that period-doubling is asymptotically independent of the form of the equation, was due to Mitchell Feigenbaum.

  3. Bryn Jones Says:

    The additional equations that came to mind when I read the list have already been suggested (F=ma, the Einstein field equations, the Dirac equation, and e^{i pi} = -1).

    It puzzles me that the equations almost all come from physics and mathematics (including statistics). The one exception comes from economics (finance). Surely there must be similarly important equations from other fields. Chemistry? (But the Schrödinger equation is very relevant in chemistry.) Biology? I can’t think of any obvious candidates.

    • Anton Garrett Says:

      Biology isn’t an intrinsically quantitative science. It took me a long time to grant it equal recognition in view of that fact, but I now most certainly regard it as such.

      • Bryn Jones Says:

        I think biology is quantitative, but in a practical numerical sense. I mean in the sense of being based on numerical data which are interpreted statistically. It may have fewer of the neat, precise analytical equations of physics. All the same there must be some fundamental equations, but I’m not familiar with them.

        Interestingly, I’m not sure what astronomical equations I’d nominate either.

        And I should have mentioned engineering as another field that should have equations of fundamental equations.

      • Bryn Jones Says:

        -> fundamental importance.

      • Anton Garrett Says:

        Anatomy is quantitative? How a cell works is quantitative?

      • Anatomy can be descriptive and aspects of it could also be quantitative. How a cell works can be quantitative – it is possible to quantify the concentrations of chemicals and how they change over time and with position in the cell. Describing numbers of cells, their characteristics and behaviours can be described statistically.

      • Anton Garrett Says:

        Yes, you need numbers in biology, but its fundamental ontology is not expressible in equations as in physics.

      • Biology certainly cannot be expressed precisely and succinctly in simple equations of the form encountered in physics and mathematics. That doesn’t mean that equations do not, and cannot, form the basis of biology. I imagine that a complex mix of processes in living systems removes the simplicity.

        There may, however, be some neat, simple equations that might be added to the list here, but my lack of knowledge of biology beyond a basic level prevents me from being able to name any.

      • Anton Garrett Says:

        From GCSE right up to the research level the science of biology is not taught as a set of equations.

      • Yes, absolutely. I don’t remember any equations during my O-level biology lessons, which is a pity.

      • Anton Garrett Says:

        What equations would you have liked to see?

      • telescoper Says:

        Michaelis-Menten equation for enzyme kinetics? Or perhaps the Hardy-Weinberg law for genetics?

      • Bryn Jones Says:

        It might have been possible to study simple models for the growth of populations, concentrations of chemicals within cells, genetic inheritance, growth rates of organisms, or similar things. The biology I was taught was almost entirely descriptive, requiring remembering details, names and some processes.

      • Anton Garrett Says:

        Ultimately it’s about pattern recognition, as is physics. But the patterns found in physics are usefully expressed in equations. I don’t believe that of the patterns occurring in biology.

    • Bryn Jones Says:

      I suspect the main difference between biology and physics is that living systems experience a number of different, complex processes occurring simultaneously, which makes it difficult to represent biology with neat equations.

      • Anton Garrett Says:

        That’s true but is not the deepest difference. Those processes go on in an environment, namely the organism; and how did that come to be as it is?

  4. Adrian Burd Says:

    I suspect numerous scientists including Murray, Hardy, Weinberg,
    Mangel, Baldocchi, Allman, Rhodes (some have been mentioned before, but I’m just looking at some of the spines of books on my bookshelf) would disagree with Anton’s (mis)characterization of
    biology. Such a view may have been arguable 70-100 years ago, but not today.

    • Anton Garrett Says:

      You think that biology is as fundamentally quantitative enterprise as physics? I think it isn’t – but is not any the less a great science for that. I reckon most biologists would agree. How do you account for the fact that biology isn’t taught as fundamentally quantitative at any level, from GCSE to the research frontier? How is the way an organism works quantitative, or a cell? There are processes going on in each that you can characterise by numbers, of course, but no equations that “sum it all up” as there are in physics. I respectfully suggest that there is a bias effect on this blog which is largely contributed to by physicists.

      • Adrian Burd Says:

        Your new statement is: “You think that biology is as fundamentally quantitative enterprise as physics?”

        Your original statement, to which I was responding, was “Biology isn’t an intrinsically quantitative science.”

        Not quite the same thing. Your original statement is an absolute statement about the nature of biological science. Your new statement is a comparative one.

        “How do you account for the fact that biology isn’t taught as fundamentally quantitative at any level, from GCSE to the research frontier? ”

        The first part of this statement is probably correct (it’s been a very very long time since I took such exams). The reason, as I see it, (and speaking as someone who comes from a physical science background and now works in an interdisciplinary subject that requires me to know biology and use quantitative skills to solve and model problems in biology) is that, unlike physics where there are a relatively small number of different phenomena to understand, in biology, there’s a vast amount of knowledge that has to be covered beforehand. This is something that has, and continues to, trip up some physicists who make staggeringly ignorant statements about biological systems. Other physicists do take the time to learn something about biology, and some of those make significant contributions to our understanding of how biological systems work — much of it in the cellular, genetic, and ecological realms, though physiology and anatomy also require a good quantitative background (….just pulling off a book from my bookshelves….for example “Physicochemical and Environmental Plant Physiology” by Park Nobel; or “Mathematical Models in Agriculture: Quantitative Methods for the Plant, Animal, and Ecological Sciences” by John Thornley and J France to add to the list of authors I gave previously).

        The second part of your above quote is really not true at all. Is all cutting edge biological research highly quantitative? No. Can you get away with doing a PhD in biology without quantitative skills? Maybe, but it’s quite unlikely.

        The days where biologists were descriptive scientists are long gone. And computationally, it’s the biologists who are pushing the bounds of computational systems. I’ve been involved in a project where the hard core biologists on the team swamped our University System supercomputer, not the physicists running their (by comparison) simple full 3D eddy resolving hydrodynamic models of the tropical Atlantic.

        All of this shows that modern biology is an intrinsically quantitative science, again, countering your initial statement.

      • Anton Garrett Says:

        For all you say, there are no equations that “sum it all up” in biology as there are in physics; a point that is as germane to my first statement as my second.

  5. Most obvious omission from physics seems to be Newton’s laws of mechanics.

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