ML is the next tetraethyl lead in fuel except much worse. Pls no.

For me, it's symplectic geometry.

can you draw all assortments of 3d objects into 2d planes(paper) precisely by the lines only, using the object's and view properties like size, length, angle, or coordinates? something like screen space coordinates/vector but to deal with all sorts of objects of all kinds? almost like guessing the right kind of lines to do for certain forms based on its position, angles?

Testing tags echo "Code with syntax highlighting!"; Inline $\text{LATEX}$ $${\int }_a^b{x}^{2}dx$$

>>71 What's so cool about symplectic geometry?

Need a book recommendation on elementary probability, must have lots of exercises and solutions. Things like: "If 13 cards are drawn from a standard deck, what is the probability they are all red?"

Damn, I don't this thread and board to die, but both seem dead af :c

>>112 Yes, this thread is dead.

In 'Foundations Of Geometry' (attached) there is an axiom which states that for any two points on a straight line, there is a point on the line between them, and then on page 11 it states a theorem that there are infinitely many points between any two points, presumably the proof of this is that between any distinct points A and B the axiom is applied to this to get another point, C, which lies between A and B, then the axiom is applied to the points A and C to get D lying between them (D lies between A and C) and so on forever. This proves that there are countably infinite points between any two as the points are proven one after another (D is proven after C and so on) so there is a bijection between them and the natural numbers (C to 1, D to 2, and so on), however normally a straight line is thought to be represented by an interval of real numbers with the length of that interval being the line's length and every distinct point in that interval is a distinct point on the line, but there are uncountably infinite elements in that interval and so uncountably many points on a straight line, yet the proof only gives countably infinite points. The book doesn't even give a proof, only stating it is 'obvious' and the proof is one I came up with, so am I missing something or is this a serious deficiency in the Foundations Of Geometry?

In page 37 of David Hilbert's book called Foundations Of Geometry (attached), there are dotted lines which are not (sandwiched) between any two other lines and the book says Pascal's Theorem guarantees that these lines are parallel but I don't understand that because Pascal's Theorem, as given on page 31, says that IF two of the lines (sandwiched) between two others are parallel to each and IF the lines not (sandwiched) between any two others are parallel, THEN another pair of lines (sandwiched) between two others are parallel to each other, so as far as I understand it Pascal's Theorem cannot be used to prove the outer lines parallel since that is one of the premises it operates on. Can anyone please explain this to me?

>>116 Could you argue something like this >make a bijection between the points on the line segment and the natural numbers >between any two points there is another point >but we already have a bijection, so this point cannot be part of the countable list of points >therefore there must be an uncountable infinity of points Like a diagonal cut

>>387 No wait that won't work because then that would prove that the rationals are uncountable, which is false. Maybe David Hilbert wasn't as smart as he thought he was.

>>116 >and what exactly do you men by a line? If we are operating with a line in euclidean space as suggested by the axiom of parallels (finite dimensional real vector space) then a line is implicitly representing a bijection of the real numbers. More rigorously we can think of a line as being the set of all points a*x where a is a real number and x is a non-zero vector in that space (If we want to think of a line as connecting two points then this vector would be the vector based at one point which points to the other). But that trivially surjective since we defined the output of that as a line and it is similarly injective since if a*x = b*x then a*x-b*x = 0 or (a-b)*x=0 using the distributive laws of vector spaces. Then we can use the lack of zero divisors on the real numbers (if a*b=0 then either a=0 or b=0) to say that since we required x to be nonzero that a-b=0 or a=b. And two infinite sets are defined to have the same cardinality if there exists a bijection between them. From the text of the book however it never states the cardinality of the infinities and simply says unlimited.

>>387 This proof fails since while there is a point between any two points of the bijection you never showed that it wasn't already included elsewhere. In hilberts proof he constructed a number that specifically wasn't in the bijection by shifting one of the terms of the infinite decimal expansion for each element of the bijection.

>>393 So basically a diagonal cut maneuver.

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