A Case for Finitism in Engineering
Mathematics is often done using Real numbers. The Reals are continuous: given two different Real numbers, there is not only another Real between the two, but an infinite number of them.
All measurements that we make are discrete. My ruler has 1 centimeter wide marks, thus I can only distinguish multiples of 1 cm. Yet… looking at the ruler I can imagine a “2.5 cm” that lies in between 2 and 3, then a 2.25 cm and 2.75 cm, then… Tada: the Real numbers! They are a natural progression from assumption “there are more possible measurements to distinguish than my measuring device allows.”
Yet: the Planck time and length (1e-43 seconds, 1.6e-35 meters) are lower bounds on our ability to measure. We cannot distinguish differences below these limits. As agents that are embodied in the Universe there is an unbridgeable gap between “what can be measured” and “what could be.” Is there a such a gap between “what is measured (by an observer)” and “what is (according to that observer)”? No! An agent can only know what it measures, what it observes and acts upon.
Finitism, finite math, is a rejection of the Real numbers and mathematics that relies on the infinite and un-measured. In place of the Reals we may use Rational numbers that can be represented exactly. Instead of algorithms that take infinite space or sequences that [take infinite timev, we put bounds on all operations and are explicit on the number of steps taken and resources consumed. To prove something we construct it.
As Humberto R. Maturana reminds us, “Anything said is said by an observer.” All the symbols we produce, all measurements and actions, are discrete and finite. Engineering is construction of real things, of other objects that are embedded in the Universe with us. As infinities cannot be constructed, their usage obscures what we may create and how we may create it.
We are intelligences that are embodied in the Universe. To embrace this is to embrace our limitations, our finitism.
An Alternate Case for Finitism
- All transmission of information occurs over a physical medium.
- All physical mediums induce noise in the transmission of information, at least through random thermal motion.
- Noise creates a finite upper bound on amount of information a channel may carry.
Thus: only finite things may be said.