what are we studying when we are doing mathematics?

map and territory

Consider map and territory 1 2:

  • Science interprets territory, formally, to compose representational maps
  • Engineering composes (or shapes) territory, by applying map to circumstances, through action
  • Mathematics, central to science and engineering, formalises cartography and navigation of map and territory, and more: mathematics charts the form and characteristics 3 of the nexus 4 between map and territory – fundamentals of constraint, structure and behaviour 5, of interpretation, representation, and cognition 6

alignment

Consider that:

  • Maps evolved to align with territory – like a promise made in the past, of some realisable future circumstance
  • A good map aligns with territory sufficiently well that one result is future intersection with a desired state of territory; or in the inverse, avoidance 7
  • Maps relate prior circumstance, through the present, to future circumstance – maps draw a line from a, through b, to c

precision

The (circumstantial) precision of mathematical maps is curious 8.

Where does this precision come from?

Consider “a fabric of cognition”, woven in material substrate of the embodied brain:

  • If the fabric of cognition is a whiteboard, the precision of mathematical alignment seems almost mysterious. Did humans invent this precision? If so, what of other animals?
  • If the fabric of cognition is more like a canvas, with weave, and mathematics charts relative form and characteristics of the weave (of the material substrate of cognition), then alignment of maps to territory might simply be a result of the constraints, form and characteristics of cognitive substrate (territory) mirroring those of universal phenomena (territory) more broadly

universality

We appear to have evolved the means to consider all universal phenomena, even those never faced by our ancestors (that we might have adapted to understand) 9.

Evolution is not known for such magical foresight, nor redundant capability.

What if universal interpretation, representation and cognition was the result of composition and composed complexity descended from the same simple primitives? Such that, the simplest form of each to evolve, was manifest in a minimal, universally applicable fabric?

what are we studying when we are doing mathematics?

Perhaps, charting and navigating the fabric of cognition, at the nexus between map and territory.

10

  1. All maps are abstract – cognition is operational map, so also abstract. ↩︎

  2. We live by maps, they instruct us, they are the nature of knowing – and while we are well served by remembering that a map is not the territory it represents – in any moment, a sufficiently accurate map is circumstantially synonymous with territory. ↩︎

  3. Math is a map. When we do math, that map is operational. We are charting and navigating abstract environments. ↩︎

  4. By “nexus between map and territory” i’m referring to the system (incl. Interpretation, representation and cognition) (to be described in detail from first principles) ↩︎

  5. All maps reduce to constraint, structure and behaviour – which is threshold through relative structure (with behaviour modelled as structural change through time). This includes the logical grain of persistence, and is fundamental to a first principles account of (interpretation, representation and) cognition, and requires detailed work through, I suspect. ↩︎

  6. A tired system (or conditional sequential mechanisms) ↩︎

  7. Alignment and intersection is concluded by meat on reactant, or some other outcome - we need territory to survive, and all things considered, even flat-wrong abstractions which don’t kill us can result in desirable intersection with territory (gestures around at humans). Maps, and cognition, are a tiered affair – “close enough for another tier to resolve” is generally sufficient. ↩︎

  8. By precision I’m referring to QM (QM is touted as profoundly precise, over some metric), but any precise abstraction fits. Maps point to fragments, or fractions of territory. Yes I agree, it’s all rather fuzzy! but the fuzziness is hidden (because can you imagine the mess we’d be in if it wasn’t?) ↩︎

  9. Universality precedes social interactions, necessarily. And can be demonstrated by a first principles account. My method includes minimal-viability, because evolution, but it also tracks through time. ↩︎

  10. Hey visitor! let me know what you’re interested in and i’ll see what I can add. ↩︎