Thread · HN Continuity

I'm pretty sure the vast majority of people understand linear algebra etc, if they were taught that the reason why linear algebra works is that algebra (algebra over a field) are an approximate isomorphism to physical reality.

From that perspective you do not need contrived examples like using function composition as alternative operator.

It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.

If you don't understand this then it's because you don't understand linear algebra.

Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.

Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.

Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.

They will also notice that marbles under physical composition follow the same rules.

They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.

They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger

They invented an isomorphism.

Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.

And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.

Edit: if it's not clear, any physical composition operator can form an algebra over a field,

In a mechanical calculator, the gear movement is the composition operator

In a computer the adder is an electron based composition operator

Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state