I found in it:

p. 203, column 1, line 13-15:

It says “The sum above will be unchanged if we add a multiple

of N to r , so we now care only about the values

of f at points of the form n/N,” but, I think, the conclusion

“so we now care only about the values

of f at points of the form n/N” does not seem to be

a consequence of the statement

“The sum above will be unchanged if we add a multiple

of N to r.” ]]>

evidently satisfies ; thus *almost-complex structure* is present. Writing the dynamical equation in Hamiltonian form for a suitable tangent vector , symplectic form , and Hamiltonian function , and moreover defining as a Riemannian curve-length, then exposes the complete Kählerian triple of metric, symplectic, and complex structures.

The starting problem “give a power series for trigonometric functions” then can be appreciated (by beginning students) as a natural meeting-ground for classical analysis, classical algebra, and their various modern syntheses that include algebraic geometry and algebraic dynamics. That’s why it’s such an illuminating topic for students to study!

]]>Show this by establishing that the (unit) radius and (unit) velocity of the integral curve both are unchanging; this can be done both formally and empirically (the latter by numerical integration) without reference to the power series.

Then as the second step, verify by explicit substitution that the power series solves this equation. QED.

]]>I read that in the 30th Anniversary LMS popular lectures you had talked about the problem of tiling a chessboard with dominoes in relation to automated theorem proving.

I don’t know, but wouldn’t that (or some similar topic) be a great mathematical blog topic? :-)

]]>Some simple definitions and arguments in point-set topology translate effortlessly into diagram chasing computations, and perhaps this observation may hint how the human mind think in that area, and lead to an automatic theorem prover similar in spirit to the one you have written. The prover may be reasonably easy to write because diagrams involved have lots of finite spaces (in simple cases), and diagram chasing is already quite computational.

The following paper details this observation.

]]>One way, articulated in great detail by Norman Wildberger, would be to avoid the notion of angle altogether, replacing it with a potentially simpler concept – spread (cf: http://web.maths.unsw.edu.au/~norman/ and in particular http://web.maths.unsw.edu.au/~norman/papers/WrongTrig.pdf and his YouTube channel). Then elementary problems of triangle geometry (as opposed to uniform circular motion) can be treated without appeal to transcendental notions such as the sin & cos functions.

This approach is so unconventional that it likely invites the reader to immediate skepticism – that would be a mistake. Wildberger carefully and clearly elucidates his approach, and he extends it to Hyperbolic and Spherical geometry obtaining some remarkable and truly novel results (cf: Universal Hyperbolic Geometry II, KoG 14 (2010) )

]]>Call a function f: R –> R a sinoid function if and only if the following conditions are met:

1. For any circle *sector* S such that its area A(S) (which I’ll define below) is at most pi/4, f(2*A(S)) is equal to the length (which I’ll also define below) of a line segment perpendicular to one radius delimiting S going through the the end of the other radius delimiting S.

2a. f(pi-x) = f(x) for all x

2b. f(-x) = – f(x) for all x

2c. f(x + 2pi) = f(x – 2pi) = f(x) for all x

Then sin: R –> R is the unique continuous sinoid function. (Again, uniqueness is guaranteed by Gowers’ angle bisection argument).

Here is a definition of A:

Let X be the set of finite regions (or shapes) definable in our Euclidean space (whatever it is). Then A:X –> R is the function such that

1. A(x) = 1 if x is a square with one of the radii of our chosen unit circle as a side

2. A(x) = A(y) if x and y are congruent

3. A(x) = A(y) + A(z) if no point is both inside (i.e. within the boundary of) y and inside z, and the points inside x are all and only those which are either inside y or inside z or on their shared boundary.

That should give you the right function, (by Gowers’ squares argument). Now for length:

Let Y be the set of line segments and circular arcs definable in our space (strictly speaking we only need line segments for our definition of sin). Then l:Y –> R is the function such that

1. l(x) = 1 if x is a square with one of the radii of our chosen unit circle as a side

2. l(x) = l(y) if x and y are congruent

3. l(x) = l(y) + l(z) if y and z share at most an endpoint and the points on x are all and only those which are either on y or on z

4. l(x) < l(y) < l(w) + l(z) if y is an arc, x, w, z are three line segments forming a triangle such that x connects y's endpoints, and w and z are tangents to y touching y at its endpoints.

That way you wouldn’t have to assume the plane has the complete structure of R^2 at all. We know from Galois theory that you can have perfectly good Ruler and Compass spaces that are much smaller than that.

But on further reflection I guess I’d retract that: sin is a function from real numbers to real numbers, so in order for our geometrical definition of sin to even work, we do need it to be the case that the plane has something like the structure of R^2.

But if you wanted to keep the geometric bits of the argument pristinely Euclidean, an alternative would be to take the geometric definition of sin to be a partial definition of sin, as follows:

* Call a function f: R –> R a sinoid function if and only if, for any segment S of the unit circle definable in Euclidean geometry, the following conditions are met, (here A(S) is S’s area, defined by the sort of Dedekind construction hinted at above):

1. f(A(S)) = the length of a line segment perpendicular to one radius delimiting S going through the the end of the other radius delimiting S

2. f(x+pi) = – sin x for all x, and sin(x+2pi) = sin x for all x

(That’s not very elegant, but it does the job)

* Let sin x: R –> R be the continuous sinoid function

Your argument about angle bisection guarantees that this last step returns a well defined function.

]]>It also occurs to me that there is a way of defining angle that requires a limiting argument, but not integration. We can bisect angles, and that makes it easy to define all multiples of by a dyadic rational, and then taking limits gives us the rest. I think it should be reasonably straightforward to use that definition and plug it in.

]]>