Calculus is more natural than arithmetic
The position of math in the universe has been a constant pointless dispute among "philosophers", and it's still an active pointless discussion among tech-monster types and Intelligent Design types.

Plato and Aristotle staked out the two sides, which haven't changed and haven't settled the issue in 2000 years. A question that can't be settled isn't a real question. It's just a way for "philosophers" to justify their grants.

A possibly more fruitful subdispute arises in math teacher circles, as well as the AI and ID types.

Is calculus more important than arithmetic?

My teacherly position is that the fine details of calculus are only needed by engineers, and even engineers don't really carry on the type of thinking that happens in a calculus class. They just use lookup tables for the results of specific equations, and multiply the results by constants as appropriate. Before computers the lookup tables were paper, now they're silicon. Same thing.

= = = = =

Here's a somewhat fresh thought, based more on neurology than teaching.

Integrals and derivatives are basic and universal parts of life. Arithmetic is NOT a basic part of life. Arithmetic is an overlaid skill invented late in history by humans.

Every neuron is a derivative. We process only changes, not constants. As the signal moves up through higher layers in the ganglia and cortex, it is differentiated several more times. Some feature detectors respond to linear upsweeps in frequency. Second derivative. Others respond only to accelerated upsweeps. Third derivative.

Many neurons are also integrators. They pick up changes from a network of input dendrites, and sum up the total of the changes over time. Some of the changes are additive (excitatory), some subtractive (inhibitory). The neuron will typically emit a series of pulses when its integral reaches a threshold set by yet another field of inputs. These pulses may feed into higher-level integrators that sum up the readings from the first layer.

And the whole system is a constantly running infinite set of balanced equations. Negative feedback balances or counteracts inputs to maintain a steady state of temperature or muscle tension or anxiety or weight.

By contrast, simple arithmetic is rare in the nervous system. There are a few integrators specialized as counters, but there's no explicit addition or multiplication of counting numbers.

Labels: Equipoise, Real World Math

  ¶ 4:57 PM