Q: Why does math work so well at modeling the world around us?

Mathematician: One important thing to realize about mathematics is that it was primarily created for practical purposes. For example, numbers were likely used in the beginning to count possessions, multiplication for trade, and geometry to measure plots of land (or some similar purposes). Mathematicians and scientists use math to model the world by constructing mathematical objects that capture important properties of physical things (while ignoring those properties that are not relevant for the investigation). Hence, it isn’t as though math just happens to work well for analyzing the world we live in, rather, it was specifically designed for that purpose. If our original mathematical objects had failed to capture important properties of real objects, they surely would have been discarded and replaced with ones that would be more useful. To give one example, if the operation of addition did not so closely model so many physical phenomena (e.g. if I have two objects in one group and I combine them with three objects in another group, then my new group has five objects, which is mimicked by 2+3=5) then it might not be considered a basic mathematical operation like it is today.

Once the basic objects of math were introduced (for their practical uses), it was then possible for people to generalize these objects, find connections between them, and prove theorems about them. For example, once we have integers (for counting) we can ask the question whether there is any largest integer. Once we have addition, we can ask the question whether a + (b + c) = (a + b) + c. Once we have division, we can introduce the idea of prime numbers. Once we have exponents and real numbers, we can introduce polynomials, and attempts to find the roots of polynomials will inevitably lead to the introduction of imaginary numbers. Hence, from the basic useful mathematical objects, a whole complicated structure follows which contains many new ideas relating to or emanating from the original ones.

Long after most of the basic objects of math were created, attempts were made to axiomatize the subject (i.e. provide a small set of basic axioms from which the rest of math can be derived), but math was not developed from these axioms. Quite to the contrary, these axioms were developed from the already existing useful mathematical system, and hence the axioms somehow inherently have built into them the usefulness of the entire mathematical structure. By altering these axioms mathematicians can (and have) developed different versions of mathematics. One thing that is special about the version of mathematics that we are used to is that it allows for creating a staggering variety of useful models. When the basic axioms are fundamentally altered, this is not necessarily the case.

A more difficult question than why math works so well at modeling the world, is the question of why math that is developed for one purpose (or, sometimes no purpose at all except theoretical interest) ends up being so useful for other purposes, but this is a subject that deserves a post of its own.

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11 Responses to Q: Why does math work so well at modeling the world around us?

  1. Julie says:

    Follow up question:

    So if math is derived from concrete things (i.e. counting chickens), at what point did it we make it abstract? In teaching middle school, 7th graders in particular, we are always talking about the transition from being a very concrete thinker to understanding more abstract and complex ideas. Especially in math, this seems to be about the times kids learn algebra: slopes of lines, etc. Certainly calculus is abstract.

    Scott also wants you to model a 6-dimensional hyper cube. Since this isn’t a real thing, how did we get from modeling the world around us to modeling things that we imagine?

  2. The Mathematician Mathematician says:

    Hi, thanks for the questions. When we model the world using math, we don’t model all aspects of the world. For example, if we want to apply math to money, we generally don’t care about the shape of the currency or how much it weights. For many problems we may only be interested in the fact that money can be pooled together (e.g. if I have one dollar and you give me two dollars, I now have three dollars). The appropriate mathematical objects to model this situation are (positive) numbers, and the appropriate operation to use on these numbers is addition. Numbers together with addition capture the properties of money that are of interest in this scenario. However, as soon as we introduce numbers and addition, we can begin thinking about these objects in the abstract, forgetting that we only introduced them to study money. For example, we can ask questions like “is there a biggest number?” and “is it possible to write every number as the sum of two other numbers” and does a+b = b+a? The physical scenario leads us to a mathematical model which leads us to theoretical questions about non-existent objects. We should therefore expect that mathematical theory began a very long time ago, and as a matter of fact, we know that the ancient Greeks studied the properties of numbers more than 2500 years ago. What is particularly interesting is that quite often the study of theoretical properties of mathematical objects leads to practical applications (which in some cases were not even intended). To give a hypothetical example, theoretical study of the number Pi and it’s properties could have allowed ancient people to more accurately measure the area of circular things (since we know that the area of a circular is Pi times the radius squares) which could have real world uses. Once we have a mathematical model of a situation, we can manipulate it using the techniques of math, which sometimes will tell us things we never knew about the situation.

  3. The Physicist Physicist says:

    1) Observe the world. 2) Make up some simple math to model it. 3) Follow the pattern. 4) Curse the gods.
    Generally things get derailed somewhere in step 3. For example, from the pattern of vertexes, lines, squares, cubes, hypercubes, hyper-hypercubes (?), that we can see clearly in 1, 2, and 3 dimensions you can predict (with impunity!) that a 6-d hypercube will have: 64 vertexes, 192 lines, 284 squares, 208 cubes, 72 hypercubes, 12 hyper-hypercubes, and 1 hyper-hyper-hypercube. The reason that this is “abstract” as opposed to “concrete” is that we’ll never be able to actually build one of these things. The reason a cube isn’t abstract is that we can build it.
    Also, I don’t think calculus is all that abstract… Velocity, y’alls!

  4. Chris says:

    I was very pleased to find this little post in response to an impulsive Google search for “why does math work”. I look forward to the followup post alluded to in the final paragraph!

  5. melanie says:

    why does numerology work! There is no need to be a “psycic or intuitive to read the numbers. They tell all purely by the number.

  6. Steven Ross says:

    What is the mathamatical function between a logical mathematical proof and an event which occurs in the universe? If I travel down a straight highway at 30 miles per hour for two hours, I will travel 60 miles. This must be so logically. It works in the universe. Why does logic in concepts match what happens in the universe? What is the mathematical map between concepts and the physical universe?

  7. The Physicist The Physicist says:

    The math we use (the applied math anyway) is just a bunch of logical structures that imitate the behavior of things we see in the world around us. Asking why math matches what happens in the universe is akin to asking why hammers are so good at driving nails: they do what they’re designed to do, and if they don’t we find something better.
    There’s no particular function for translating back and forth between the physical and mathematical worlds, but there is a philosophical approach.

  8. Physics Major says:

    The idea that math models the world is within itself an abstraction. The world is infinitely complex, more complex than math currently allows to show. For example, describing molecular and atomic interactions of big molecules. However, instead of tackling the overbearing complexity, we look for patterns. With these patterns we then search for other patterns until something tangible is found, hence from addition came multiplication, and from multiplication and from multiplication whole numbered exponents. This also addresses the point of whether math is created or discovered. Math is certainly created, we did not decide the one and one make two, we only decided to name it such. Or, in more generality, math is discovered, but notation is created

  9. Rajesh Swarnkar says:

    I guess it only does – currently. My belief is that reality would be far more complex than what is in front of our eyes if we ever knew it full. Math is just current attempt to describe that.

  10. jing says:

    why is math used to model real world situations?

  11. Ygg says:

    This Is Such A Good Article To read

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