My reading, through university, was heavily weighted with philosophy, I like to think by choice, and among those widely considered important philosophers, whom I read, was Ludwig Wittgenstein. I’d also been biased towards psychology thanks to its interest in myth-making and dreams. Philosophers do not neglect dreaming, by day or by night, as an aspect of life.
His thesis was that signs gain their meaning not by pointing or labeling, but through usage patterns. Captioning and indexing, ways of naming, involve elaborate grammars, sets of rules.
Signs serve as tools or instruments, as implements, as artifacts. We observe them in use, and then may adopt and adapt, evolving our own form of life or private language.
Except the idea of a truly private language is all nonsense. Seeing language and life forms as coterminous helps us remember body language, the patterns in architecture, and music. We may lose any sense of a boundary, once language includes all the things and “mirroring through representation” is no longer required.
Wittgenstein was a master at crafting examples and leading us, step by step, into a way of considering the activity of using language. Perhaps stone slabs and heavy equipment were involved. Were we in some battle or other contest of weapons? Context matters so much that it induces perceptions and keeps our thinking from becoming unglued. A shift in context may be hard to pinpoint, in terms of what just shifted. Wittgenstein’s Philosophical Investigations anticipates gestalt psychology. Meaning derives from use, yet “use” has roots of its own, in timing and rhythm. New meanings sometimes “pop” into view (occur suddenly).
Wittgenstein’s examples in the mathematical arena have been branded unsuitable given currently raging debates, although many have jumped to his defense. He was perhaps insufficiently Cantorian for his day. He lived on the cusp before computer science, with Alan Turing one of his students. Did his approach anticipate computer languages, in drawing attention to “meaning through use” (not “naming”)?
Yes, the XYZ coordinate system is a “language game” with clear rules. Then Quadrays comes along as another “language game” in the same domain, likewise with clear rules, yet with alternative notions of what a “basis vector” might be.
The four quadrays fan out into a set of four rays, four sectors each bounded by three of the four, the four basis vectors for all linear transformations. Any point in space may be represented as (a, b, c, 0) vis-a-vis (0,0,0,0) at the origin. One place or slot is always 0 as one basis vector is dormant per each sector i.e. (0, a, b, c) would locate a point in space in the quadrant opposite (1, 0, 0, 0).
Maths consists of this “foam of namespaces” wherein common words, such as “vector”, gain their spin from the local rules (field of meaning), yet may also be perturbed by more external usage patterns.
However, a single global “essence” corresponding to the “true meaning” of a word is an illusion and delusion very common among philosophers. The idea of words cleanly encapsulating and pointing definitively to some “essence” that is their meaning, is their pot of gold at the end of a rainbow, their “unobtanium” — a superstition in other words.
Partial recognition, some “family resemblance” (another Wittgensteinian concept), proves sufficient when it comes to cohesion.
We no longer suffer from the same ontological requirements, about what “must” underpin our language, in order for it to make sense. Not as much as we had imagined.
Want to see Quadrays in action? Click here for a Jupyter Notebook treatment.