Financial derivatives are another example of application of Digital Constructivism, or DC. Financial derivatives are to be considered here as a good example of a quantum system, and of non-continuity.
In financial derivatives, there IS a natural, smallest unit of payment, such as (In English law) a peppercorn. One cannot provide less consideration than a legal peppercorn, in payment. That is the quantum of it, legally and naturally established.
Smoothing is used to get nice properties in mathematics (not just in finance), as a field of reals. However, integers mod p (Z/Zp) are a field as well, and much more suited to the problem, bringing Galois fields to describe financial derivatives, with a digital topology. No more smoothing and its cost.
Because of cryptography applications, Galois fields are also much faster. One can use it as a hardware drop-down library (Intel) in present computers.
What is your qualified opinion, in both counts?
From another thread, and relevant here, to illustrate how multiplication can change,
"I ran a C++ program (code below) that compared the default multiplication method with multiplication as extended addition, obtaining an improvement of about 3,000%. According to [1], it should be even faster with Galois fields.
double timei, timed;
clock_t START;
int x;
double y;
const double M = 10000000;
int k = 0;
START=clock();
while(k
A number in a Galois field (see previous posts here) can have any, fixed precision one desires! It does not have to be 0, 1, or even 8.
It can be as high as desired, or seems possible naively. We are talking about Z/Zp. It can be 2, or 27, or 271, ore 2718, or 27182, or 2718281828459045235360287.
And one will calculate using the ONLY four operations of arithmetic (+-x ÷), they can be performed EXACTLY and are CLOSED in Z/Zp. We don't need the reals anymore, with numbers that cannot be even counted (the irrationals). And they are MUCH, MUCH faster (check with the x of two primes, like 13 and 47, in Reals and in Galois fields).
And then one will do derivatives (two - and one ÷ operation) and anti-derivatives (used for integrals) as arithmetic, like one's own computer does. And, the calculations are MUCH faster! One will be happy -- infinitesimals or continuity did NOT even come up! The computer can do algebra too.
And one now can go further, and faster. One realizes that physics, engineering, CS, and other sciences are not based on error, uncertainty, noise... but certainty. One no longer accept the Heisenberg principle... THE QUANTUM COMPUTER IS BORN TODAY! FIAT!
From another thread, and relevant here, to illustrate how multiplication can change,
"I ran a C++ program (code below) that compared the default multiplication method with multiplication as extended addition, obtaining an improvement of about 3,000%. According to [1], it should be even faster with Galois fields.
double timei, timed;
clock_t START;
int x;
double y;
const double M = 10000000;
int k = 0;
START=clock();
while(k
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