Object-oriented versus expression-oriented mathematics | Arithmetic and Geometry Math Foundations 77

Twentieth century mathematics has been object oriented. Twenty-first century mathematics, if it gets its act together, will be much more expression oriented. Here we describe the distinction by studying the key example of the unit circle.

This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.

Video Content:
00:00 Intro to distinction between object and expression oriented maths
6:23 The unit circle
11:54 Expression-oriented approach
14:18 A Projective bipolynumber
16:20 Interpretation A
23:49 Interpretation B
29:58 Simple case of a Projective polynumber
33:45 The second most circle
37:41 Interpretation C
42:45 Polar / pole correspondence

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Here are the Insights into Mathematics Playlists:

[peterhi]

Wildberger, your work continues to be first rate. You may be the finest expositor of mathematics of whom I am aware today. Well done, man. Hardy.

[hideakipage]

I really enjoy your videos. I know that you are a critic of modern math. The refreshing thing is that you understand both sides of the argument and offer an alternative. I don't like orthodoxy. However, we need rational criticism to move away from the status quo. Please keep up the good work.

[pfrpadilla]

You are the only person that seems to be talking exactly what I am trying to explain to my students. I have taught science for 17 years and I have found that students are not understanding some things because of how we are explaining it, and I talk about objects and expressions, math as a language that describes objects, actions, and properties of those objects and among them. So when we see a math expressions interpretation depends on the type of objects and properties and actions definitions, so if you want to discuss I gently and humbly raise my hand.

[ThatHippyPerson]

I am a hobbies. The plumpton 322 interest me. I appreciate all you have taught me. You are amazing. Thank you

[DanPartelly]

Really cool how simple you can extend Int B to present partial derivatives and their directional generalization.

[StevePatterson]

Lovely video. I would be interested in hearing more elaboration about the pointillist circle. That's how I think about geometric objects. The pointillist circle you've drawn doesn't look continuous because there are too few points--in fact, if you look closely enough at the "smooth" circle above it, you'd find it's composed of a finite number of bits of ink on your whiteboard. Therefore, it must be that "smooth" objects can actually be constructed out of discrete points. Smoothness, in this perspective, is a matter of having fine enough resolution.

From my understanding of history, I gather that the Epicureans thought similarly about geometric objects--they rejected the notion of Euclidean, infinitely divisible space. Are you aware of any schools of thought which approach mathematics this way? And could you say more about your hesitations towards this perspective?

[stilingiceland]

Thanks for the very clear/concise lecture. I felt the golden age of math and physics would be 18 century ~early 20 century. The mordern math is somewhat too abstract. Just my cents.

[prajnaprajna]

On Only (N)
x^3=[x(x+1)/2]^2 - [(x-1)x/2]^2. 
Defined the function f;
 f(z, x, y)=[z(z+1)/2]^2 - { [[x(x+1)/2]^2+[y(y+1)/2]^2 }.
So
f(z-1, x-1, y-1)=[z(z-1)/2]^2 - { [[x(x-1)/2]^2+[y(y-1)/2]^2 }.
Impossible at same time both
 f(z, x, y)  =0 and f(z-1, x-1, y-1)==0.
If   
z^3=x^3+y^3 or z^n=x^n+y^n.
ADIEU.