The difference between the algebra of values (numbers) and the algebra of functions.

[studiot]

This thread is aimed particularly at @hoola  and @bangstrom

 

It took from the mid 1600s to the late 1800s to appreciate the full significance of this difference and even today it is rarely brought out in courses.

When schoolboys move up from the algebra of values eg solve x+3 = 8  to the calculus they are introduced to 'functions'.

But few, if any, are taught that functions have an algebra of their own.

And this algebra is different (more complicated) from the very simple algebra of values they have become accustomed to.

The function and its derivative (short for derived function) have a particular property that they exist (mathematically) whether we use all the function or not.

Sometimes the function and its derivative are 'out of phase'  the sine and cosine functions provide a good example, but both exist for all x.

 

Now I see several current and past thread where members have faled to appreciate the implications and significance of this when applied in the physical world.

For example entanglement, non commutative results, the fact that the solutions of the wave equation extend over all space, again whther we use them or not.

 

This last example is the reason other members have been telling bangstrom that there is no first and last in such experiments because the wave functions are not points and should not be treated as such.