Essays and notes on geometry & information.
How much information a message carries and how meaningful that message is to a person are different questions. Mathematical communication theory can measure specific aspects of uncertainty and representation without measuring human memory, context, or value.
Mathematics is where symbolic reasoning most dramatically escapes immediate experience and then returns with predictions about…
The same position can be represented in several coordinate systems. The place does not change; the rules of description do. That distinction is part of what makes geometry interesting to me.
Computers represent many observations of nature digitally, but the representation and the observed world are not the same thing. We choose which distinctions to preserve, how finely to measure, and what to discard.
Geometry describes relations of position, distance, angle, curvature and transformation. A coordinate system makes some relationships convenient to express, but it does not create the underlying…
A coordinate system is deliberately chosen. Physical spacetime and biological form impose constraints that may not fit everyday Euclidean…
Geometry is a discipline of invariant relationships: descriptions can change while structure…
Software architecture also lives by abstraction: a good model compresses many cases while preserving the invariants that…
A mathematically correct model can still encode the wrong physical assumptions. Certainty inside a formal system is not the same as empirical certainty about the…
Numbers sit at the border between discovery and invention: we create notation, yet formal relations often reveal consequences we did not…
When similar patterns appear at very different scales, we naturally look for a connection. That instinct can begin an investigation, but visual similarity does not prove that the same mechanism produced both structures.
Data engineering teaches the same lesson brutally: a value without schema, unit and context may be precise and still be…
The success of numerical description does not prove that physical objects are literally numbers. Representation and ontology are different…
Architects make similar choices when selecting schemas, reference frames and abstractions: representations can simplify reasoning without becoming the thing…
Greek mathematical traditions tied proof, ratio and geometry to philosophical questions; the historical Pythagoras is particularly hard to reconstruct because no securely attested writings…
Numbers let us count, compare, order and express ratios. Measurement maps a physical situation onto numbers, which means the unit, operation and interpretation are part of the…
Euclidean geometry built a powerful deductive system; non-Euclidean geometries later showed that apparently obvious spatial assumptions can be alternatives rather than…
Mathematics constructs formal structures from definitions, relations and inference. In science its power comes from mapping aspects of physical systems onto structures from which consequences can be…
Ancient Pythagorean traditions connected number, ratio, music and cosmic order, although later sources make it difficult to separate the historical Pythagoras from doctrines attributed to his…