via Peggy Holman today http://www.opencirclecompany.com

Tolstoy

The key story:
Imagine you are in a dark room and you start feeling your way around it. Eventually you will learn all there is to know about the room and its contents. You will find the light switch, read all the books, watch all the videos etc. Imagine also that you come across a locked door to which you do not have a key and which you are unable to kick down. Most scientific disciplines would offer solutions at this point. Engineers would try and build a camera to fit through the key hole. Physicists would try to garner as much info as possible from the light seeping through the key-hole. Ditto chemists, who would attempt to understand the chemical composition of the universe beyond by analyzing the atoms that travel through the key-hole.

Maths outdoes all the other sciences. It would try and find an object (better still invent an object) that adheres to the stipulations of permanence and correspondence; as well as, most importantly, fits through the key-hole. Correspondence requires that there be a one-to-one correspondence between any given point on the new object and myself (or whoever is stuck in the original room). Think of the one-to-one correspondence between the pimple on the face and cape town on the map on the other side of the sheet of A4 paper (the example we discussed earlier). Its imperative that the correspondence is one-to-one, i.e. pimple corresponds only to cape town and vica versa! If cape town can correspond to pimple as well as left earlobe, the whole thing breaks down. Permanence requires that whatever mathematical (or otherwise in our analogy here) rules of operation defy the person in the room, also apply and defy the new object. And thus the mathematician succeeds where the others faltered: he actually gets to travel through the keyhole himself and explore another room and this can go on for ever (or untill the whole universe has been exhausted).

Basically, all mathematical tools one learns about in school or university are such objects. Logarythms, algorythms, trigonometry, calculus etc etc.

As an easy example of permanence, think of the ancients to whom the very concept of a negative number was absurd. How can you have minus 27 chickens??? To them, numbers started with one and that was that. They went through the keyhole as it were to another room when the admited the negative realm to their numbers. They realized that the positive number room was just a small room within a much larger room that contained the negative numbers as well. Permanence dictated this since all the rules that define positive integers (the commutative, associative and distributive laws for the sake of simplicity) also admit negative numbers. So even though the concept of a negative number seemed alien to the ancients at first, they had to accept that positive numbers were but a mere subset of a much larger family - a family that through the ages (and with the help of correspondence and permanence) grew to include ratios, surds, irrational numbers, transcendental or non-algebraic numbers, imaginary numbers, complex cojugates and all sorts of matrix-based contraptions that are frankly beyond me.

Hope this helps Natalie and see you soon.

Balint