| ™ |
Tom Veatch
Flames of inspiration often leave smoke signals behind.From mine, these.
| Mean Thoughts | Thoughts about the Pythagorean means: the average, the geometric mean, and the hyperbolic mean. This is surprisingly deep for so simple a topic. Understanding at least 2/3 of it is part of your path to wealth and meaningfulness. |
| Negative Dimensionality | Abstracting geometry. Hilbert's axiomatization of geometry, full of redundancy, led me to a generalization which makes geometric dimensionality a characteristic that can be counted up (as in point to line to plane to space, etc.) and down (space to plane to line to point: etc.) Geometries, by intersecting, create lower-dimension geometries; for example two intersecting 2D planes create a 1D line. Geometries, by projecting, create higher-dimension geometries; for example, two 0D points project a 1D line. But if there is no upper limit, perhaps there is also no lower limit. The idea that a geometry might have negative dimensionality seems absurd, considered within the assumptions of spatial thinking, yet it derives from the same less-redundant axiom set as the geometries we understand. Suggestions for intuition and use of this idea are also given. |
| A More General Theory of the Syllogism | Abstracting logic. Aristotle's
list of syllogisms missed half of them; there's nothing to them
(H!); and we can do better
without. Still it is pretty fun and cool, considering this was the intellectual pinnacle of humanity for 2000 years, and plus I'd say this is not a bad introduction to "term logic", and might be suggested reading for students of computer science, philosophy, classics, and/or math. |
| Bliss Theory: Emotion in General | On a mathematical represention of emotion, with decomposed functions including Identification which turns out to have a central role. |
| Math as Language | Underlying intuition reads out as discrete expression. |
| Math Tutor | Learn your times table. |
| Neural Networks + Fuzzy Logic + Space | A careful, accessible introduction to neural networks
assuming only high school algebra and a little geometry and
differentiation. NNs are defined mathematically, along with how to
run them, how to train them (by the usual gradient descent), how
to train them better (so I suppose: using 'Newton-Raphson', which
really ought to kill!). I also discuss how to understand the
training algorithm's implicit reasoning about the adjustments it
decides to make; I share an interpretation that backpropagation is
like an Anti-Dunning-Kruger learning system (and therefore morally
superior to most men?). Then I give a whole Fuzzy Logic
re-interpretation of NNs, along with suggestions on how to enhance
their logical reasoning capabilities. I tried the wikipedia page,
and got so frustrated I wrote my own introduction. So yes, I
suggest reading this if you want to really understand neural
networks, and if your other resources have made it seem
inscrutable. It's a few pages of actual math, yes, but all the
steps are laid out: no leaps! It's not short, but you don't have to be
a math major to follow along. I encourage your
study here if you are interested in really knowing how neural
nets work. Also this adds Fuzzy Logic to neural networks, including how to train them. Finally this goes into Space Representing Neural Networks so robots can represent space, or humans' representation of space can be understood better. Three months of work is in here. |
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