Discrete Mathematics

This is really good. You see this correspondence between fractions in the Stern Brocot tree and continued fractions very clearly. See also his previous lecture on this.

You could make a great text book for discrete maths and functional programming out of Norman's videos. Just about everything he does is programmable because of constructivist, finitist stance. See Doron Zeilberger and Persi Diaconis on Probability and Mathematical Knowledge.

Subscribe to Insights into Mathematics.

And you can look at matrix multiplication as circuits:

42:06 So  it turns out that the category of matrices is a full subcategory of the category of finite dimensional vector spaces over a field. This means that you can define finite dimensional vector spaces over some field and then you have matrices (1:06:50). See the lectures below. So does this mean that the natural numbers can be modelled as matrices? And could you then use rational matrices like those that Wildberger uses above as elements of the underlying field for the vector spaces? And then the next stage is to define tensors as derivatives of matrices so that you can start enumerating models of the kind Chantal Roth constructs? See Pierre Agostini - The Genesis of an Attosecond Pulse Train.

See also Emily Riehl, Justin Clarke-Doane and Simon DeDeo on Meaning and Truth in Mathematics and Brendan Fong and David Spivak - Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

Subscribe to Richard Southwell. 

 

Subscribe to Frederic Latremoliere PhD.

 

See also Gabriele Carcassi on The Correspondence Between Quantum and Classical Mechanics.

 

Subscribe to Topos Institute.

Juan Aguilera on a modern view of mathematical Platonism (and a whole lot of other -isms, I had no idea about!)

49:08 He drew a picture of the Universe (of Sets) and has an interesting idea about dark matter! See Alison Gopnik - The Evolution of Human Intelligences.

 

52:10 Question about existence and proof in physics v. mathematics. (53:13) Applying the scientific method and postulating axioms: the axiom of infinity is a good example: it is independent in ZFC and leads to all the foregoing. Something similar happens with the Axiom of Choice and infinite dimensional vector spaces. See basis theorem.

 

55:33 So is it fair to say that it is the axioms that are real? The thing is that the consistency of the axioms is a property of the whole set, not any individual axiom. 

 

56:25 On anti-platonism, see Norman Wildberger's view of the rationals as the objects that cast shadows across the plane of Gaussian integers or whatever the points really are, ... 

If you treat formal systems as experiments in this way then it seems you do introduce a kind of dynamics into the process of mathematics. The dynamics takes place on hyper-surfaces where proofs in different axiomatisations can be connected by homotopy paths (see e.g. Emily Riehl on Univalent Foundations). So maybe this is an answer to the question Michael Levin asked Roger Penrose via Curt Jaimungal (see Another Curt Jaimungal Interview With Roger Penrose). See 1:22:34.

1:15:13 Question about the size of the set of all ordinals. I think this is the Burali-Forti paradox.

 

Subscribe to Michael Levin.

 

One comment reads: My discrete mathematics education was in the same course as functional programming. As a result, I don't remember discrete mathematics nor functional programming.

Subscribe to The Math Sorceror.