Tag Archives: math

AI as ‘Journey Thief’… Continued

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In yesterday’s post, ‘Journey Thief‘ I focused on artificial intelligence stealing the learning journey of students. In a comment, Karen Flello said,

“I used to ask students how they would know if they liked a subject (topic, activity) if they had never tried it. Then we talked about how “in the real world” we use things in ways we never expected. And maybe they are right – maybe they will not use a particular skill again. But if they never had exposure, they would not have the opportunity to find out what they would use (or like, or want to know more about). And I agree that AI can rob us of the learning in service of producing something that earns a grade. We have not earned it – we have not learned it – and that is the sad part. AI should be a tool to help us. But “help” means we, in fact, are doing something in the first place. And prompting some other thing / person to do something for us does nothing for developing imagination, skill, or resilience. We just get better at asking something / someone else to do it for us. A whole new kind of dependence.”

And I replied,

“Great to hear from you Karen,
That idea of lack of developing imagination, and also dependence on AI to do the work (and ultimately the thinking) is one of the biggest fears I have with AI. I’m simultaneously excited about AI taking humanity to new insights and new heights, while also deeply concerned about how much it can steal from our own learning journeys.”

This got me thinking about recent developments in Mathematics. In the past couple weeks AI has been solving a significant number of problems and conjectures that have stumped the best mathematical minds for entire lifetimes. This is incredibly destabilizing for the field.  Imagine being one of the foremost experts on a mathematical problem, having dedicated years of schooling and even more years afterwards to solving an seemingly unsolvable problem… Then having an answer dumped out of a computer along with answers to several more difficult problems that others in your field have also committed their lives to solving. You’ve moved from expert seeking answers to expert checking the work of an AI. You’ve moved from researcher and publisher to AI grader. You’ve watched your life work solved, not by someone smarter than you but by a machine that will only get smarter still.

Math is the first of these high knowledge fields to be ‘solved’ by AI, what’s next? It seems physics is not far behind.

These insights and solutions will do a lot to improve the understanding of these fields, but for how long? Does a current PHD student in Math who saw a future in solving one of these problems now continue on with their future plans? Does a newly graduated undergrad even decide to continue on with a Math PHD? Does a high school student find excitement in delving into Math when they know that they’ll never solve a problem that AI won’t beat them to?

What happens to these fields in 20 years when all the joy of discovery is stolen by AI? Who in 20 years is going to have the desire to enter this field? I asked Google who the top mathematicians are today and it came up with a list including Terence Tao, Sir Andrew Wiles, Grigori Perelman, Marina Viazovsk, Sir Martin Hairier, and Edward Witten. Who replaces them in a generation when all the problems they seek to solve are already solved by AI?

We already live in a world where only a handful of people really understand how computer chips are made and could help to rebuild the technology if all of the knowledge was lost today. Imagine how few that number will be in 20 years when robots are building AI designed chips to further improve AI. Imagine how many fewer will get into fields like Math and Physics, much less computer programming, when the most they can do is get a tertiary understanding of the field.

Is AI going to rob us of the next Einstein or Newton? Will anyone in 50 years understand the universe as much as the brightest humans today do? …When AI is doing all the work for us?

Like Karen said, “We have not earned it – we have not learned it – and that is the sad part. AI should be a tool to help us. But “help” means we, in fact, are doing something in the first place.”

What work will be left in these fields for humans to do?

___

 

The math behind experiences

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I wrote a 3-Part West Coast Trail series (1, 2, 3) after returning home from the trip. My buddy Dave and I, ‘The Two Daves’, have had some amazing experiences in 6 weeks. St Mark’s Summit, Everesting the Crunch, Sea to Sky, Tofino, and the West Coast Trail. I feel blessed to have had the opportunity to experience these things with Dave… and many others who have joined and supported us.

My visits to many places and gratitude towards many people has me thinking… The experiences we have are built on a simple equation:

Experiences = Places + People

We need both. Places that take us out of our usual routines, and people to share those places with.

It’s a simple equation, and yet it’s what brings us joy in our lives.

Discovered versus Invented Math

Is Math discovered or invented?

The conclusion I come to is BOTH!*

This is an oversimplification but I think it makes the point:

There is no such thing as 1 divided by 2. I can have 1/2 of 2 apples, that’s what we would call discovered math. I have 2 apples, I spilt them into 2 groups of one, and give you 1, and I have one left. We can do this easily. But if I cut a single apple into halves, then I have invented Math. I can’t grow a half of an apple, and I can’t cut an apple perfectly into halves.

Another way to look at this is that if I take 7 equal length 1-foot rulers and put them in a straight line, I have a 7 foot length that is made up of 7 single units. However, if I decide to make this length a single unit and I divide it into 7 parts, 1/7=0.142857 repeating.

There is math that we have discovered because it is discoverable in nature, and then there is math that is invented to make sense of ideas that we invent, like fractions of a whole. Using the 7 units of 1 divided by 7, versus 7 divided by one, is a clear example of this. A perfect 1/2 of an apple is another.

But it’s a bit more complicated and confusing than this simple explanation. This doesn’t mean that any irrational number is necessarily invented. π (pi) is discoverable and not invented. The golden ratio, Φ (phi), is also discoverable. The square root of 2 on the hypotenuse of a 1×1 square is an invention we sought to make sense of an imaginary line dissecting a square.

So, some Math is discoverable and some math is invented (capitalization is my own emphasis).

– – –

*Note: I’ve come to this conclusion through discussions with Joe Truss on the nature of our universe and the premise that ‘We Live in a Tetraverse’.

Map of Mathematics

I came across this Map of Mathematics in a TikTok where people working in the field of Mathematics were all sharing how their fields intersected different parts of the map. Here is the video:

I love how the concepts and fields of study connect and interconnect.

The the video starts with a ‘stitch’ of a Tiktok user asking, “Did everyone know that Mathematicians are, like, discovering new Math?”

This plays into a recent post in which I state that Math is ‘Discovered more than invented‘. There is so much more to discover, and I believe, thanks to my conversations with Joe Truss, that geometry is the foundation of Math. But math is a hard conceptual language with which to understand the geometry… and if we study the geometry we can understand the universe without having to do the hard math. Oh, and if Mathematicians better understood the geometry they could discover new Math a little faster.

In the Map of Mathematics, Geometry is the key.

Discovered more than invented

Is Math invented or is it discovered? Is Math a human, or at least biological construct that helps mammals understand the world we live in? Or does it exist inherently in the universe and is it revealed to us through curiosity and scientific discovery?

I’ve shared my fascination in playing, learning, and discovering secrets held in Geometry (here and here) with Joe Truss.

Recently I watched a video that quoted Galileo Galilei:

“Mathematics is the language with which God has written the universe.”

With everything Joe and I have been playing with, I thought of an adaptation of this quote:

The universe is written in the language of geometry, and Mathematics allows us to measure it.

When I shared this with Joe, he tweaked it:

The universe is written in the language of geometry, and Mathematics allows us to take its measure.

I like this because it speaks to the idea of Math being discovered, of it being inherent because it is present in the geometry. It is Geometry that is the underlying language. Math is the alphabet, the universe is written in geometry. This is why Pi pops up where you don’t expect it; why the Fibonacci sequence is prevalent in nature; why we can land a ship on the moon and Mars; and even why our religious symbols come out of geometry.

The challenge we have is that we don’t fully understand the geometry, sometimes rather than discovering the true essence of the mathematics, we have to invent approximations that help us make sense of universe. Understand the geometry and maybe we can remove infinities and irrational numbers in our calculations… Understand the geometry, and the math becomes more accurate. Understand the geometry and maybe we can unify our theories that currently seem to contradict themselves.

Discover the geometry and we discover the math behind it, no longer needing to invent approximations of the math to translate the geometry into a world of calculations and numbers that don’t quite fit with they universe we are trying to describe. When we discover the geometry, we no longer needs to invent Math, the geometry allows us to take its measure… And in fact we can explain the universe with the geometry, and then we don’t have to be a physicist or mathematician to understand it.

Playing with geometry

The image below is a blue cube octahedron with each triangular face forming a tetra-octa-tetra pattern coming out from the center.

This is a tetra-octa-tetra, with two yellow tetrahedrons on opposite sides of an octahedron:

I’m fascinated by these shapes thanks to my uncle, Joseph Truss, who has been playing with these shapes for decades, and who believes the fundamental building blocks of our universe is the tetrahedron. We live in a ‘Tetraverse’.

But the geometry he describes goes far beyond this. He intuitively understands the underlying structure of the universe through geometry in a way that mathematical physicists understand it through math.

Watch this 28 minute movie, Hacking Reality. It gets to some of the topics Joe and I discuss. I really think he’s on to something some of the smartest physicists in the world are still trying to figure out… but he’s done it purely by playing with geometry, and visually/intuitively making sense of how shapes are formed and come together.

Fun with Math

Take ANY number, subtract the digits in that number, and the answer is divisible by 9.

14… 14 – 1 – 4 = 9

25… 25 – 2 – 5 = 18

36… 36 – 3 – 6 = 27

49… 49 – 4 – 9 = 36

1,234… 1,234 – 1 – 2 – 3 – 4 = 1,224

And you don’t need your calculator to figure out if 1224 is divisible by 9, it is. How do I know? Add 1 + 2 + 2 + 4 together and if it’s 9 or divisible by 9, then the number is also divisible by 9.

7,692… 7,692 – 7 – 6 – 9 – 2 = 7,668

And then 7 + 6 + 6 + 8 = 27 which is divisible by 9 so 7,668 is divisible by 9 (= 852).

I love Math.

The answer doesn’t seem right

Back in December I shared that we really don’t understand exponential growth. Well I’m about to share a question with an answer that will boggle your mind. All I ask is that you make a guess at the answer before looking at it.

Imagine we get every single human being on earth to play a ‘winner play on’ game of rock-paper-scissor. First, let’s round up the population to 8 billion. And let’s assume every one of them can play.

A game of Rock-Paper-Scissors with 8,000,000,000 people.

So the first round will be 4 billion pairs of people, with 4 billion losing, and 4 billion winning and moving on to the next round. The question is:

How many rounds of rock-paper-scissors will the winning person have to win consecutively in order to be the world champion?

Guess.

I’ll give you a hint: it’s less than 100.

What do you think?

Here is the answer.

Well, what if we added another 1/2 billion people? That would be 500 million more players, or 8.5 billion total. The answer would still be the same!

Hard to believe. It’s so difficult to wrap your mind around needing this many games to whittle 8 billion people down to one winner. It just doesn’t seem right… but it is.

The real numbers

In the last week and a half I’m aware of 18 people whom I personally know have caught covid-19, likely the Omicron variant. But of these 18, (half of which are 2 families), I think only one of them is reported. Others sat on 811 for more than 2 hours before giving up on the call. Others have mild symptoms, and here in BC they are asking people with mild symptoms not to get tested, because testing locations are being overwhelmed.

Extrapolating on these numbers, even if I overestimated the amount of cases in BC to be 1/3 reported (that would be 6/18 rather than 1/18), then the real number of new covid cases is a massive number.

This is surprisingly a good thing. First of all, our hospitals are not overwhelmed with serious cases. Secondly, at this rate of infection, we are moving quickly to heard immunity.

Well, that’s my hope anyway. I am only working with my own anecdotal evidence, and coronavirus has kicked my predictions in the butt more than once. Still, it’s good to look forward and see a little light at the end of the tunnel.

Maybe by spring everything will be open and masks will be optional. Maybe. Still, it feels good to have a little optimism right now. Even if we really don’t know the real numbers, I think they are looking good… Time will tell.

Doing STEM

‘Doing STEM’ or ‘Doing STEAM’… there is a saying, “Put lipstick on a pig, and it’s still a pig.”

I don’t want this to sound like a rant, and I don’t want to knock teachers for trying to do STEM projects. I do want to say that if 5 years ago a teacher did a project with kids where they broke them into groups and had them assemble a limited number of straws and a specific length of tape into the tallest possible tower, and if they do it again today it isn’t suddenly a STEM project.

Now, if that same lesson included teaching geometry and/or structural integrity; or if students had to design it such that their design had to have a function such as offices or apartments; or hold a weighted satellite dish; or if it had to factor in wind resistance (such as a blow dryer at close range); or if they had to model their design first and estimate the height they will achieve… if there was some thinking, designing, modelling, or estimating that was required before or even during the build process, well then it’s looking more like STEM.

Hands-on does not equal STEM. Building something does not equal STEM. Group challenges does not equal STEM. Meaningful integration of cross-curricular concepts, where problem solving requires thinking in more than one subject area relating to Science, Technology, Engineering, and Math is STEM. It doesn’t have to check all the boxes, but it should include thoughtful integration of at least a couple of these.

It’s about making the cross-curricular connections explicit, or at least thinking through how the outcomes and expectations relate to these connections. It’s about developing competencies in the areas of STEM and not just doing a project that looks like STEM.