H0kay, try this little exercise. Draw a diagram of a crossing from two points to two points, that is, draw a planar representation of a braid group generator. Now if you draw the same diagram rotated by 90°, it's equivalent to the inverse of the original generator. Another way to get the inverse is a reflection through a horizontal or a vertical line. The original is invariant under rotations of 180°, but the points (say you label them with a, b, c, d) are permuted. Suppose you want to use these switching elements to make a set of nonintersecting paths, so the only important function is the permutation of paths, not points. That's a way to quotient a braid group--make the two orientations of each crossing generator equivalent; in effect you choose two inputs and two outputs, and put the crossing in a black box. The only other kind of operations you allow are merging of two tracks (paths), and branching one track into two.
Roger Antonsen has a wonderful presentation on viewing "modules" (4 triangles) from different perspectives. Actually his whole presentation is very entertaining and informative; https://www.ted.com/talks/roger_antonsen_math_is_the_hidden_secret_to_understanding_the_world
Obviously without numbers and measurement, mathematics would not exist in our world. Modern mathematics to me, is like a refinement of ideas, the process is ongoing, mathematics "researches" the patterns we keep finding in numbers. Sometimes, perhaps because we don't need it until we do type of thing, some patterns aren't noticed. I know it's a bit parochial, but the Rubik's cube was around long before Erno Rubik "invented" it. For some reason, the notion of permuting parts of a cube, seeing what happens to the global symmetry when it gets changed locally, didn't seem to be interesting, and you can object that a solid cube can't have its edges or vertices permuted. Please Register or Log in to view the hidden image! If somebody had ignored all the objections and just done it "abstractly", who knows? Fermat, or DesCartes, maybe? Gauss? None of them saw the opportunity. And a five year old can build a larger cube out of smaller cubes. But that aside, or never mind how long young humans have played with blocks (no, when I was five I didn't think about permutations and combinations of cubes either), mathematics is like that video says, about finding patterns and then switching perspective to see how the context might change. Mathematics ignores the objections and just does it anyway. The anthropology of numbers, if a certain author is correct, has a history of development that supposedly follows an as-needed curve. Nobody looked at the permutation space of a stack of ordinary cubes, because nobody needed to know at the time. So the time it took academia to adopt the square root of a negative number is on this curve (supposedly!). I'm a fan of the cube, but not a fan of speed-solving contests. To me, that's only an example of an algorithmic solution that runs on some computer.
Here's an exercise in context-dependence. Suppose you draw a square using geometry. In fact the drawing represents diagrammatically the abstract geometric object. Now what if you want to square this square? Can you take the square of a square? In the geometric context, does it mean you scale the sides and the area? Well obviously if you say the sides are all length 1, then the square of your geometric object is the same object. And "the square of" means the operation of squaring (multiplying the object by "itself"), distinct from "a square", so maybe avoid the confusion and dereference the terminology. The square of two squares is a larger square, but you have to have two squares, you need to be able to freely add squares. The "larger square" of two geometric squares (or regular 4-gons) is a combination of four squares, but there are two ways to combine two squares. What seems to be needed is a "tiling" rule so the combinations are in fact squares, meaning they are aligned edgewise with no overlaps.
If I may make an extraneous observation: There is a difference between "purpose" (intelligence) and "filling a need" (necessity and sufficiency) https://en.wikipedia.org/wiki/Necessity_and_sufficiency In context of logical numbers, it is obvious that any consistent symbolic representation of mathematical functions are allowable. They are descriptive and informative about recognizing conditions, values, functions, and emerging natural needs, but is nowhere "written" in the universe. It merely functions in accordance to the natural imperatives of "Necessity and Sufficiency", i.e. the emergence of a mathematical "Equation" (a+b=c). This is how I understand Bohm. Mathematically implicated "physical relationships" and an explicated emergent "sum totals" of values, i.e. patterns in our reality, in a greater permittive but probabilistic condition or field.
I like Kauffman's idea that algebra is something like a recipe: start with (a) geometry, then think about changing something, introduce a line of points, maybe think about embedding the geometric object, or set of them, on a cylinder or a cone. What changes for instance, if you try to embed a square tiling on a cone? And so on. A square graph is more general because it can be drawn more than one way, but it retains some of the rules of a Euclidean tiling if you connect a pair of "graph tiles" together in the same way, which means removing an edge from one of the pair of graphs. This is a standard graph-theoretic "move", like say, row reduction is in matrix algebra
M'kay, so the diagrams of a 1-point shift 'generator', which I'll repost: Please Register or Log in to view the hidden image! . . . wrap n lines around a cylinder. The rule is that if you have n points, you need n compositions of one of the above to wrap all n lines so they each 'move' through 2π (radians, if you insist). What does u look like when n = 1? It's just a line wrapped once around C (the cylinder). Two lines, when n=2, are wrapped by a 2-point shift, so they appear to cross over each other. If you connect the trailing lines with say, a dashed line, it crosses n-1 other lines in the diagram. So this baby, with its vertical edges identified, is useful, but easier to visualise with low n.
What is algebraic structure? It's something many algebraic objects have a visualisation of. A Klein bottle has an algebra visualised by an animation, the abstract function that changes a cylinder into the bottle is different to a function that changes a cylinder into a torus. Both morphisms involve stretching and bending so cannot be Euclidean. A structure with a lot more algebra is the Klein quartic. Here it is as a hyperbolic tiling: Please Register or Log in to view the hidden image! I posted a still image of a video because of the blue 'tile' in the centre. It's a composition of 7 triangles, each triangle is colored twice so the coloring divides the triangle into two smaller right triangles. What if you can remove one of the 7? You have a hexagon, a plane tile. Here's the youtube vid: Since 7 is a prime number, and 6 is the product of two primes instead, a 7-gon is hyperbolic and a 6-gon is flat. Remove another tile and you have a curved surface again a 5-gon, three of these almost make an icosahedron if you glue them together the right way. Or as they say, if you identify pairs of edges the right way.
Another vid from the same guy: (It uses the same "pair of pants" decomposition, see if you can line up the colored edges in the first "leg", which has two pink boundaries that end up as a line on the surface of the first half-torus. Note the decomposition leaves behind parts of each colored boundary in the tiling . . .)
So here's an idea although it might not fly (it could well be defective in the sense the hyperbolic tiling becomes unstable, although that could also be interesting . . .). The blue tile in the still image: can you cut around it and fold it so it looks like it has 6 triangles? Say you allow creasing and fold one of the 7 so it's vertical, but so there are still 7 contiguous triangles? I think so, but you're not supposed to change the topology--a crease in an otherwise flat (at least locally) surface, is a discontinuity. So after changing the blue tile (breaking the symmetry!), you're supposed to glue it back, and there are 7 edges to re-connect. You see that you will need to reduce (diagramatically) this by 1, by folding 1/7 of one of the adjacent tiles, so one triangle of 7 is vertically projected out-of-plane (the hyperbolic surface of the quartic is here represented as a planar diagram, so all this transformation does is keep the (representation of the) surface flat, in two perpendicular directions. Now the curvature of each (no longer hyperbolic) tile is determined by the amount of 'lift' from one side of the vertical triangle to the other . . .
Well, you can try to make a hexagon out of paper, then cut along a 'radius line', and see if you can add a 7th equilateral triangle. It doesn't fly because you need to preserve the partitioning of 360° (or just the interval (0, 2π)). With a hexagon as a composition of 6 equilateral triangles, the angles at the centre are all 2π/6, and this needs to change to 2π/7 if you add another triangle. You keep the other angle though, so going around the centre (along the boundary of a 7-gon) means going around 2π + 2π/7. So this removal or addition of a triangle as a transformation is going to mean a bit more than what I can squeeze out of paper. But paper doesn't stretch, it bends. If you roll a sheet of paper into a cylindrical spiral, bending it in a direction other than the one it's already curved in isn't possible. So I need a mathematical model of a triangle that I can change one angle of, without making the surface . . . unstable. Maybe just forget the surface and focus on the edges, so a change of angle will also change the lengths of a pair of sides. It should be calculable. In other words, the hyperbolic tiling is a planar representation of a set of flat regions where the angles add up to 2π, which is where the seven 'radius lines' of the 7-gon meet. In between, at least along the boundaries of the n-gons, the white lines (the white and blue boundaries in the image below, are dual to each other as graphs of a function), there is an angle of 2π + 2π/7 around each flat centre where 7 blue lines meet. Please Register or Log in to view the hidden image!
So to answer the question "what is an algebra?", perhaps try to answer the question "what isn't an algebra?". Algebras are possibly as ubiquitous as vector spaces. Hell, the language I'm typing here is algebraic. It isn't a rigorous algebra, it has more freedom than a strict set of rules would impose on it. It isn't a regular language, the grammar isn't context-free, etc. But here are some examples from Wikipedia (not such a bad reference point, at least at the kick-off).
This is why Tegmark posits that the deeper you look the more mathematical everything becomes, until at the smallest scales there are only patterns of mathematical values and functions. Humans assigned symbolic numbers to the relative values and symbolic operants to the functions. In the case of Algebra humans assigned symbolic letters to undefined values.
I wanted to address the apparent discontinuity between mathematics (say, a science of numbers), and physics (a science of measurement). Although there's more to mathematics than just numbers, and more to physics than just measurement, the discontinuity I cite is that between exactness and approximation. For instance I can divide a circle into exactly n parts, the parts don't have to be equal. If I have a 10 dollar note, then I have a thing whose value is exactly 10 dollars, regardless of exchange rates, inflation etc. In physics exactness and measurement do not commute. In physics, the best we can do is build measuring devices which have a certain precision (even if we don't know its value), then we can say that any number of measurements of the same thing should give a statistical result which approaches the limit of precision. That is, we have exact numbers, exact formulas, but inexact measurement. Although I can count an exact number of sheep, can I rely on my counting method? Should I build an automatic sheep-counter, and how accurately will it count sheep? So that numbers as exact discrete objects exist, sheep are discrete, but counting them accurately is somehow a different thing, it introduces the notion of an approximate value which might be good enough, but how do I know it is?
That is not quite correct, IMO. A 10 dollar note may be worth 1 penny in actual physical worth, but it represents a symbolic value, it is a promissory note, a symbolic object. If you are not satisfied with the accuracy of human measurements of natural values and functions, invent more accurate measuring devices. The universe is not responsible for our limitations. What we observe or measure has absolutely no effect on the exact intrinsic value of a physical object and the exact functional interactions. We may be fooled, but nature is never confused or in doubt. It provides an exact permittive and restrictive framework for all extant values and functions. If our calculations are wrong, nature will advise us by yielding a different answer than we projected. Natura artis magistra.
What is "actual physical worth"? You mean the material the note is made from? Accuracy and precision have nothing to do with whether I find it satisfactory. We invent more accurate measuring devices continually. Why bother doing that, if the universe of human measurements is always going to be approximate?
Yes, the paper itself is worthless, it is the promise of payment that has the value. It depends on the scale of the measurement, no? At Planck scale, do we need exactness other than for theoretical calculus? But if I buy a pound of sugar, I'd like to know that I will get a good approximation of a pound for the promissory note (a check) for the exact amount of money.
Yes, I can go with the second part of your sentence at least. The piece of paper has a monetary value, which it "inherits" from a bunch of symbols, pictures, etc. One of these is a string of Arabic numerals, maybe some letters--the serial number. But "worthless" is relative. Perhaps you should qualify it in terms of the symbolic monetary value, compared to which it is pretty much value-less. On the other hand, it's what the symbols appear on, its physical value in that context is quite different, yes? Or alternatively, remove the "worthless" paper from a 10 dollar bill, and see if the monetary value changes . . . Sorry about that sounding a bit sarcastic. Mathematicians do that kind of thing with abstract(ly valued) structures. Treat a 10 dollar note like a mathematical structure and maybe you can subtract something like the paper, maybe all the symbols and other images are held together by some other structure, or maybe you get a little pile of green dust (if it's US currency). This isn't just mucking about either (albeit, mathematics doesn't need to have any physical objects like a piece of decorated paper), for instance the question: if the Euclidean plane is a set of points, what holds it together?
That's easy, I can write a legal IOU on a paper napkin. The 10 dollar bill is a means of exchange, it has an exchange value. It is merely a promissory not. Wiki. This implies that any exchange of money for any purpose involves a quid pro quo. Exchange of value for something of equal value. Of course, the SCOTUS has declared that a 10 dollar bill is also a form of speech, which I find so contrived it is shameful. Only in a Capitalist country would money be considered speech. I did not know that speech equalled purchasing power Wiki.
