Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: philosophy

  • Maths is not Science

    Let me say it again a little more emphatically: Maths is NOT Science. The major difference I want to focus on here is the concept of truth. Things are true in Maths, but they are not in Science.

    Let me explain. As far as I can tell, Science is about coming up with models that explain how things work which match closely to what we observe and allow us to make accurate predictions for what might happen next. No model is technically ever “true”, it simply does a super good job at making predictions and matching our observations. There is always the possibility that someone might come up with a better model in the future.

    This is why a very accurate idea accepted by all scientists is only called a “theory”, rather than a “truth”. The word theory doesn’t diminish the decades or even centuries of work that has gone into it or diminish the great job it does of matching the universe we see. It just highlights the fact that the aim of science is to come up with ideas that work, not necessarily ideas that are “true”.

    The “laws” you learn at school are in fact descriptions of observation, which is why they are true – the only things that are true in Science are the actual observations we make ourselves (and even then these are subject to some error).

    (Any scientist reading this, feel free to slap me for simplifying it all too much!)

    On the other hand, Maths is all about truth. A mathematician tries to find out what is true and what is not true, given a set of starting rules (and some of them try to work backwards to find a set of starting rules that will MAKE things true). And once something in Maths is proven true or false, then it always was true or false and it always will be true or false. The sum of the proper divisors of 6 will always be 6, the number pi will always be irrational with no repeating pattern to its digits, an angle of 30 degrees will always be impossible to trisect with only a ruler and compass, and two ovoids in a finite projective space of even order who share all of their tangents will always meet in an odd number of points. No-one is going to come up with a new idea in the future that changes the truth of these things.

    Of course, people may come up with new types of numbers or new types of geometries, or new sets of starting rules. They may decide that there is a better way of writing the maths notation, or that something else is better for solving problems in the “real world”. But that still won’t change the truth of all the maths before it. If you start with the same set of rules you will always get the same results. That’s what maths is about.

    So, don’t try to find a repeating pattern to the digits of pi or prove that it is in fact possible to trisect any angle! Because Maths is NOT Science – things in Maths are true forever.

  • Vector philosophy

    What is a vector?

    When students first meet vectors they are pretty much told that vectors are arrows. They move arrange the arrows head-to-tail to add them, and they lengthen the arrows when they multiply them by numbers. Sometimes the vector is represented using coordinates, but they are told that this is shorthand for the arrow that goes from the origin to that point. Lovely.

    And then, in first year maths at uni, they are told that vectors are a list of coordinates. To add them, you add the coordinates, and to multiply them by a number, you multiply each coordinate. You create sets of vectors using equations, which are represented as coloured regions in space – a point is coloured in if its coordinates satisfy the equation. The only way to make sense of this is to think of your set of coordinates as a point, not an arrow. Oh dear.

    So which is it then? Are vectors points or arrows?

    There are at least four answers to this:

    1. Vectors are points, and the arrow picture is simply a way to helps us visualise what the result will be when we add them and multiply them by numbers. So when I see a set of coordinates, I think of the point in space they represent, and when I draw a set of them, I just colour in the points. And when I add them I am just using the rules for how they add to find the answer, but I can visually picture where in space the answer will be by imagining them as arrows.

    2. Some vectors are points, and other vectors are arrows, depending on context. So when you add two vectors, the first one is a location – a place to start, if you will – and the other one is an arrow – a direction to go. The result is a the place you get to by starting at the point and moving the arrow, and is of course a point again.

    3. Vectors live in two separate worlds, one where they are points, and another where they are arrows. There is a world where vectors represent locations in space and they happily find themselves inside our outside sets. But there is another world, where vectors are arrows and they happily arrange themselves head-to-tail when they want to add. When you have two points, and you want to add them, they quickly move over into arrow-world, do their adding, and then the answer comes back as a point again.

    4. Vectors are neither points nor arrows. They are mathematical objects with no innate physical reality at all. The concept of drawing a vector as a point or an arrow is simply a way to represent them on paper and it is not what they actually ARE.

    And which answer do I think is the right one? All of them, none of them – this sort of discussion is philosphy, not maths, and in philosophy there’s no right or wrong. You pick what works for you. Indeed, being able to see it from different viewpoints only enriches your understanding.

    Still, I do quite like the “separate worlds” idea.

    This comment was left on the original blog post: 

    David Roberts 23 July 2013:

    I think the main difference is that a vector as an arrow is an inherently geometric idea: you draw them, move them around the page and so on. The list of numbers is more algebraic, and more static: you can’t pick up (2,5,1) and stick it on the end of (22,7,0) like you can with those vectors represented as arrows.

    There are really different things going on when you move vectors as arrows around the page that don’t happen in the list of numbers approach. The trick is that those concepts don’t get formalised for several years after R^n is formally introduced, and are usually couched in a much more complicated and abstract setting.