What is math?
It appears that no one is quite sure. You could say it is the study of numbers, but that leaves out geometry, probability, the study of sets, and much more. Mathematics goes far beyond numbers. Many definitions of mathematics don’t even mention numbers. Wikipedia says math “…is the study of such topics as quantity, structure, space and change.”
Bertrand Russell, on the other hand, being a mathematician himself, said that mathematics is: “The subject in which we never know what we are talking about, nor whether what we are saying is true.” (Russell, Bertrand (1901), “Recent Work on the Principles of Mathematics”, International Monthly, 4.)
In other words, no one knows what math is, but they recognize it when they see it.
I would add that mathematics is the study of patterns and it is the language of science. It is a particular way of describing and understanding the natural world.
Now lets talk about maps (don’t worry, this isn’t a complete non sequitur). A map is a description of reality. A road map it is a description of roads, intersections, towns. A topographical map describes the height of the land, a survey map gives boundaries, etc. In each case it isn’t the real thing, but you can use it to navigate the real world.
I like to hike so I’ve learned to read topographical maps. I can sit in the comfort of my living room and see that the Flume Trail has a long steep section that I should avoid if the weather is bad. If you think about it, that’s quite an achievement. Lines on a 2 dimensional surface give me true information (if the map was well made) about the actual landscape.
Math can be thought of as a type of map. It uses “mathematical objects” instead of lines on a map. A mathematical object is anything that has been defined in such a way that it can be used in mathematical processes (rather circular definition, but there you have it). This includes numbers, lines, sets, matrices, etc. Where I can read a road map to get from point a to point b, a mathematician can read a formula and see a circle, or the speed of light, or how to construct a better widget.
For example, you can map out shapes using geometry. The State House dome is x circumference. Your door frame is a rectangular opening size y, so to replace the door your dog ran through you need a door to fit size y..
Geometry is relatively easy to imagine as a map since it deals with shapes, but equations can be thought of as maps as well. “Three for a dollar” is an equation written out with words. It maps out how many for how much. Odd looking equations also have real world applications. F=ma is Newton’s Second Law of Motion. In English it says that the force (F) which is acting on an object is equivalent to the mass (m) of that object multiplied by its acceleration (a).
Roughly speaking you can use that equation to discover how hard your dog hit the door by multiplying her weight by how fast she was moving. Actually, that description would make a physicist cringe, first because size does not equal mass, second, because acceleration isn’t the same as speed.
Technically, acceleration refers to change in momentum, and mass refers to the amount of matter in a given object regardless of forces working on it – gravity for example. However, here you only need a rough approximation of the force so we’ll avoid those technicalities. Once you know the force of her run you can determine just how thick your door needs to be to prevent a repeat performance.
Imagine, f=ma just mapped out your door needs. Who wouldda thunk it?
This shows how math can be used to describe and model physical reality. However, math goes a step further than a standard map. Math predicts. It’s as if your road map could continue on into unexplored areas.
For reasons no one understands, mathematics can be used to predict how physical reality works. This leads to a great debate over whether math is invented or discovered. Why is it possible to use the rules of mathematics to predict the existence of a physical object (such as various sub-atomic particles)? Its an interesting question (at least to me and many other math affectionatos), but regardless of why it works, there is no doubt that mathematics lead to real results.
What surprised me when I first looked into math is how dynamic it is. We (meaning non-mathematicians) tend to think of math as fixed. We know all there is to know and we are simply learning new uses for what we know.
Actually, that is far from the truth. Mathematical knowledge builds upon itself the same as any other scientific discipline. I had fun looking up current mathematical topics with names like: fibered simple knots, modularity of K3 surfaces and algebraic topology. I saw a prize won for “deep work on the global Gan-Gross-Prasad conjecture and their discovery of geometric interpretations for the higher derivatives of L-function in the function field case.” (http://math.mit.edu/index.php).
Henry Cohen received the 2018 Levi L. Conant Prize from the American Mathematical Society for his paper: “A Conceptual Breakthrough in Sphere Packing” (Feb 2017, Notices of the AMS). In his remarks upon accepting the award he said:” “The /E/_8 and Leech lattices are fascinating objects, and I hope readers will grow to love them as much as I do.” http://math.mit.edu/index.php
Uh huh. No wonder we’re afraid of math.
Like I said math is a different language. But that doesn’t mean we can’t understand the reasoning behind it, at least in areas which effect our lives. I have written about how compound interest works and I would say that probability and statistics are equally important areas to understand. I will write about them at a later date. Stay tuned….
Thanks for reading,
Feb 19, 2018
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