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Principles

• Every answer is an amount and a unit.
• Combine answers with shared units.
• List answers with different units.
• A right triangle is half a rectangle.
• A triangle is the sum of two right triangles.
• Four times a right triangle is the difference of two squares.

Ideas

Several of the principles can be thought of as instructions, constructions, and equations. "A = B" means that given A, you have an instruction that takes you from A to B, which yields from A the construction B, and which equates A and B. So there are three ways to think about the same relation A->B: instruction, construction, equation.

A rectangle is a ratio.

Type checking is a check of units.

Notes

Some thoughts that came up... As regards why 10 gram x 1 cm / sec is the same as 1 gram x 10 cm / sec, the "conservation of momentum" is a consequence of the symmetry in space, in that the outcomes don't depend on any particular coordinate in space, as per: https://en.wikipedia.org/wiki/Noether's_theorem Emily Noether was an inspiring woman mathematician.

"Conservation of energy" is a consequence of the symmetry in time, in that the outcomes don't depend on any particular time coordinates. So the difference between space and time may perhaps be thought of as the difference between momentum and energy.

Energy also gives an example of what "second squared" can mean. I just wonder what energy means. The Wikipedia article on energy is mystifying: https://en.wikipedia.org/wiki/Energy Mathematically, kinetic energy K = 1/2 m v**2 is the integral (by velocity) of momentum M = m v. Velocity is the expressed relationship between space and time. The higher the velocity, the more weight is placed on time, and the lower the velocity, the more weight is placed on space. Kinetic energy is what it takes to go from velocity v = 0 to v = V. Potential energy is the background energy that explains for us that total energy is conserved.

Perhaps one difference in interpreting the fourth dimension is whether time has a plus or a minus sign. In Minkowski space, used by Einstein, time and space have opposite signs: https://en.wikipedia.org/wiki/Minkowski_space Whereas in Euclidean space/time I imagine they have the same sign. I'm curious to learn more about the reason for that distinction.

Joseph, I like very much your emphasis on units. I found that helpful in learning physics and as a tutor I developed some general principles for my students:

"Every answer is an amount and a unit" (3 is not an answer but rather 3 feet, 3 seconds, etc.) But then (by sleight of hand) a number can become a unit: 3 millions, 3 sevenths, etc.

I'm wondering about the purpose of this breakdown. It's probably partly to distinguish between what we attribute to our mental world (the amounts) and to the physical world (the units). And perhaps it helps us distinguish between answers and questions. Answers are fixed and so they are "contravariant": if we divide up the units by 1000, and go from kilograms to grams, then we have to multiply the amounts by 1000. Whereas questions are not fixed and they are often phrased in terms of (1/unit) as "per unit": How many miles per hour? And if we divide up the hour into 60 minutes, and we get "per minute", then we have to divide up our amount (How many) by 60. I'm just thinking out loud.

I taught that "You combine like units", (to calculate), for example: 3 sec + 2 sec = 5 sec but 3 sec + 2 feet isn't anything (to combine) 3 million + 2 million = 5 million 3 sevenths + 2 sevenths = 5 sevenths 3 X + 2 X = 5 X but 3 X + 2 Y doesn't combine

"You list different units" (to make your answer easy to understand) the marathon was won in: 2 hours + 12 minutes + 8 seconds

You convert different units to same units in order to combine and, in general, to make it simpler to calculate.

• Simple examples that illustrate theory.
• every answer is an amount and a unit ir tt.
• combine like units
• list different units
• a right triangle is half of a rectangle
• a triangle is the sum of two right triangles
• four times a right triangle is the difference of two squares
• extending the domain
• purposes of families of functions
• How to present deep ideas in math? Using examples and games? How do games teach? Create

learning materials.

• Relate bundle concepts to amounts and units.
• How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? and to classification in topology?
• Scaling is positive flips over to negative this is discrete rotation is reflection
• Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions.
• Study homology, cohomology and the Snake lemma to explain how to express a gap.

Counting

• The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence.
• Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo?
• A) veikla kažkada prasidėjo
• B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų