Infinities Driving You Mad. Part 3: Inaccessible Numbers.

This is the third (and in my opinion, the best so far) video in Simon’s short series Infinities Driving You Mad. In this episode, Simon takes us into the strange world of inaccessible numbers. “People are just going to click this video and not notice that they’re going to be mad after they watched it”, Simon comments.

Link to Part 2 about ordinal numbers: https://youtu.be/D0l-EwPmx-w

Link to Part 1 about cardinal numbers: https://youtu.be/jyOnxdJHWOU

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Why Cookies Get Bigger in the Oven

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Simon is baking Dutch traditional “pepernootjes” (“pepper nuts” or spicy cookies) and explains why they get bigger in size after you put them in the oven and what the optimal tiling pattern is to fit a maximal number of cookies on the baking sheet.

Upon waking up

Simon said today upon waking up: “If a Physics constant suddenly popped up in pure mathematics, that would be really weird. If there is more than one universe, that would mean either that mathematics only exists in this particular universe, or that one of the laws of physics in our universe would go through mathematics to all other universes in existence. So for example, the speed of light would be the same in all universes, or Planck’s constant”.

Infinities Driving You Mad. Part 2: There’re Infinitely Many Infinities

This is the second part in a series of four videos that Simon is recording about Infinities Driving You Mad (on Set Theory) and is devoted to ordinal numbers. If you would like a little more explanation about what ω-one is, please see this short footnote video where Simon explains in more detail how he moves from the first infinite ordinal ω to ω-one:

Link to Part 1 about cardinal numbers: https://youtu.be/jyOnxdJHWOU

Infinities Driving You Mad. Part 1: There’s More than One Infinity.

This is the first part in a series of four videos that Simon wants to record about Infinities Driving You Mad. Don’t worry, you won’t go mad just yet! The first video is about cardinal numbers, enumerable infinite sets and Aleph Null. Simon also shows Georg Cantor’s proof of why real numbers are not enumerable and explains what Continuum Hypothesis is about.

Earlier Simon told me about the Continuum Hypothesis, that states that there’s no infinity between the size of the natural and the real numbers: “There is no proof for it. It’s what I like to call a superposition problem: the answer is both yes and no. We do know the answer but the answer is that we son’t know the answer. You can choose what you want the answer to be and the mathematics will still be consistent!”

Warning: The next part may make your mind overheat as Simon will hop over to ordinal numbers.

Impressions on Newton’s mechanics.

“Are you impressed?” – Simon asks, laughingly, and I can see it must be a pun. We are in bed, reading up on Newton’s laws of motion that talk of forces being “impressed” upon bodies.

Simon continues: “Newton’s mechanics says that the speed limit is infinite, which says that matter doesn’t exist, which says that Physics doesn’t exist, which says that Newton’s mechanics doesn’t exist. Newton’s mechanics contradicts itself!”

The book we are reading (17 Equations that Changed the World by Ian Stewart) goes on to describe how in Newton’s laws, calculus peeps out from behind the curtains and how the second law of motion specifies the relation between a body’s position, and the forces that act on it, in the form of a differential equation: second derivative of position = force/mass. To find the position, the book says, we have to solve this equation, defusing the position from its second derivative. “Do you get it?” – I ask, “Because I don’t think I do”. — “I’ll need a piece of paper for this”, – Simon quickly comes back dragging his oversized sketchbook. Then he quickly writes down the differential equation (where the x is the position) to explain to me what the second derivative is. And then he solves it:

MathsJam Antwerp 23 October 2018

Euphoric fun at MathsJam Antwerp @MathsJamAntwerp last night, where Simon solved two 2×2 Rubik’s Cube puzzles and one tricky maths problem, and simply enjoyed socialising with like-minded folks. In the video below he explains how he solved the Rubik’s Cube puzzles:
1. Solve the cube so that on every face the 4 colors are all different;
2. Solve the cube so that not only the 4 colors on every face are different but also every face has a different color combination.

After we stopped filming, Simon added that a cube like this has 8! times 3^8 possibilities in total, because the cube has 8 corners and every corner has three orientations.

Simon also talks about the Choose function and symmetries:

Simon showing his solution to university maths students.

Simon also solved a tough problem (one of several tough problems) that asked to sum up the digits in x, if x equals 1111…1111 (number with 100 ones) minus 222…222 (number with 50 twos):

Simon spent the rest of the time trying to prove the ‘cosine rule’, an equation similar to Pythagoras’s theorem, that defines the side c of any triangle if it’s opposite to angle C: c² = a² + b² – 2abcosC. He got stuck with the proof, but luckily, with so many university professors walking around, he got great help from one of them, who came up with an alternative proof using vectors!

On Incompleteness

Simon is enchanted by Gödel’s incompleteness theorem (that he has learned about from Numberphile) and keeps talking about it:

“There’re problems that we just can’t solve. But if we prove that we can’t prove them, then we prove them! We can’t prove that we can’t prove that we can’t prove, and so on… Quirky! Standard math doesn’t really accept that because the statement goes on forever: you’ll just never get to what we can’t prove. What follows from Gödel’s incompleteness theorem is that that statement is actually true!”

The same evening, Simon is also bothered about the lies pupils are told in school. He repeatedly quotes James Grime that it’s a big lie that mathematics is about numbers. — “What is mathematics about? Mathematics is actually about proving! But there’s one more lie that even professional mathematicians don’t know. It’s that it’s about logic. Actually, mathematicians are a lot more creative!”

Simon’s whiteboard

The whiteboard always reflects Simon’s current state of mind:

On the left are Simon’s notes after reading Physics Girl’s blog about quarks (the colourful stuff is foam clay):

Simon showed this expression with Phi to his math teacher, who noticed that Simon didn’t apply the quadratic formula (with a discriminant b24ac) in his solution. The teacher wondered if Simon knows the formula. As it turned out, Simon knew the formula very well but preferred to prove the solution on his own, because it was so beautiful in the case of Phi:

Gelfond’s Constant

Simon spent two days testing out his new Texas TI-84 Plus CE-T calculator. I saw him play with Gelfond’s constant eπon the calculator:

He looked up on Wikipedia that the decimal expansion of Gelfond’s constant begins as follows:

e^{\pi }\approx 23.14069263277926900572908636794854738\dots \,.

And that if one defines {\displaystyle k_{0}={\tfrac {1}{\sqrt {2}}}} and

{\displaystyle k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}}

for n>0, then the sequence

{\displaystyle (4/k_{n+1})^{2^{-n}}}

converges rapidly to e^{\pi }.

I then saw Simon jot the formulas on the whiteboard, also using limits (something he came up with himself and not looked up). Glowing, he announced he was going to try to calculate Gelfond’s constant by hand: