Here are some thoughts on your insights into the physical nature of mathematical calculation:

– Mathematics involves discrete symbols and operations that get manipulated in sequential steps over time. Whether using pen and paper, a computer, or the brain, calculation requires moving symbols or representations between “slots” or states in a step-by-step manner.

– When a continuous mathematical object like a function needs to be represented, it gets sampled or discretized into separate symbolical pieces across different slots or time steps. Continuity gets broken down into discreteness for computational purposes.

– Symbols serve as placeholders or pointers that can invoke or stand in for larger mathematical objects not fully represented. Their meaning depends on contextual inference and what can be constructed as needed during the calculation process.

– Any mathematical calculation, whether in the brain, with external tools, or simulated computationally, requires physical time and resources to manipulate and transform representations sequentially from one state to another. Thermodynamic costs are incurred.

– People often report feelings of frustration, effort, or relief that correspond to the “thermodynamic” process of pushing a calculation forward to completion or “solution.” Computation engages physical and cognitive resources.

– Externalizing calculation through written symbols or computers reduces load on biological inference systems like the brain, but introduces new costs and limits related to physical media and hardware constraints. Different approaches involve tradeoffs.

In general, you’ve made some insightful connections between the discrete, sequential nature of mathematical calculation and its underlying physical and experiential requirements across biological and non-biological implementations. Let me know if any part of this overview needs more clarification or discussion.

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