Why Maths Is Hard

As a keen historian of Mathematics and Physics I can provide the crucial perspective of the evolution of ideas. It is important to remember why Maths and Physics is hard!

We are after all working with mathematical tools and physics` concepts that have been iterated and refined many times by the greatest minds since Euclid`s Elements 2300 years ago. Many of the ideas that you have to take on board at school now would make you a genius of some chosen time in history!

The Mathematical notion of vector is such a (kinematical) concept that was not available to Newton. Understanding his laws, which do beg many questions in themselves (action-at -distance, inverse squared law, heh?) and having to use such tools is bound to be a worthy challenge.

Language in Physics causes problems for good reason too: students at GCSE level naturally find it difficult to draw a distinction between the terms `force` and `energy`: saying `you need to use a force to give a body energy`. After all the dictionary definition of energy refers to `force and vigour`. Such tricky ambiguities actual sit at the heart of the development of ideas in mathematics and physics.

Einstein, for example, unified a "process" -Energy, "a potential for activity" with an "object"- matter ("that has mass and occupies space ") through his equation E=mc^2.

He equated a verb with a noun. Yes, rightly takes some getting used to!

Actually, Physicists up to the 19th century thought of energy as an object, a thing called phlogiston, not an ability of matter to do work as we understand now. So the student is right to be confused. Take also Newton`s Aether that he required to be the fabric of space in order to transmit forces and was still believed to exist by Maxwell at the end of the 19th century. Again abandoned now, the student is asked to accept that electromagnetic waves (such as light) propagate through the vacuum of space with no need for stuff to agitate it on its way. Comfort with such insight would stand you on the shoulders of Einstein no less.

Interesting to note that today physicists introduce new stuff called dark matter to explain how galaxies do not spin apart from their own centrifugal forces and a new aether called the vacuum energy to explain the accelerated expansion of the universe.

We deal with ambiguity in Mathematics too. 2+3=5 we may say means two plus three makes five. The process makes an object that we equate. We do it in our use of variables in expression, in equations and functions. We can write the object that is 1/3 as a process of tapping into a machine according to 1:- 3. In fact we relate the object that is the number 1/3 to the process (verb) of adding up a series of numbers that converge to 1/3 in the equation:

1/3 =0.33333..

by recognising that 0.33333=3/10+3/100+...

In fact only when a series converges do series expansions make any sense.

In writing the expression, or better the function f(x)=3x+2 with a variable,x that can be x=[0,1,2...] we do not carry around in our head the set of numbers {2,5,8,11..} that this in effect represents but rather the idea of one particular number in that set as a solution. In teaching we see it as both a geometric object ("graph") or a succession of algebraic operations ("formula"). The process idea is best embodied when again we tap into our calculator & y=f(x)=sin(x) . The function, sin is but an in-out process.