„It is really hard to mathematical teachers today.“
„You just have to think. Everything else have WolphramAlpha.“
My first encounter with Stanford mathematician Keith Devlin was in 2013 at Moodle MOOC Introduction to Mathematical Thinking. It was great online course and I was so proud of having attended a course at Stanford (even online). Later I read some of his books, listen to his radio show. I liked one with Danica McKellar , wrote article about her and hope that she also some day visit Serbia to promote mathematics. When I heard that Keith Devlin visit Serbia 26.2.-1.3.2018. (Beograd, Kragujevac, Novi Sad) and hold several lectures I knew that I had to attend at least one lecture. And of course have a picture with such celebrity.
This is article about Keith Devlin lectures at Department for mathematics Novi Sad and Center for Science Promotion (CPN) Beograd for all those who did not have the pleasure of personally attending. Lecture from CPN you can view in Serbian on their facebook page.
Here you can find so many useful, interesting, update information which you can transform to knowledge.
Mathematics in the XXI century – CPN Beograd
All I learned during my lifetime in mathematics how to do is now on this device (smart phone). During my lifetime I struggle with equations with four variables. This device has no problem with hundreds of them and in a fraction of second will give you exact answer. And not just arithmetics. All mathemematics now free on cloud. But I am not gone out of work. We still need lot of mathematicians. But we don’t do what we used to do. The way mathematics is done is totally different from the way used to be done. Children now gonna live in the world where the perspective of mathematics is totally changed.
How mathematics help us understand the world and do things in the world.
From the point view of people this change in mathematics is good. Mathematics used to be restricted to small number of people now how to do calculations. Now almost everyone can do mathematics in that way. Teachers have a lot of problems around the world adapting to tis new way of teaching.
Trough our history mathematics is developed to solved real world problems. Geometry for measuring land. Nil delta flood every year. Trigonometry for navigation etc. You begin with some real world situation. Suppose you want to bild a house. You have some ideas and then you go to the architect. Then architect will described your idea in his (differnet) language. Than someone draw the plans (blueprint), which are eventually mathematical description. Now one can do all mathematical calculations in advance. When you do the estimations you hire the crew to bild a house. You go back to real world. And this is the way how we apply mathematics.
In this case it is not surprising that we use mathematics. But what if you start to use mathematics to study people.
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You can tell where are things gonna be in physics. In case of crimes you can’t exactly but you can increase chances to find the criminal. And vast majority applications of mathematics today outside universities is not about finding exact answers. Real world almost never have exact answers. It is about finding better answers, better solutions, more efficient solutions.
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The most interesting part were mathematics is powerful is not the shapes you can see but invisible shapes. So triangles are good, they fun, really good to begin with mathematics. And this is where mathematics become very powerful. It reveals invisible patterns.You can represent the pattern even though they are invisible. I wrote the book: The language of mathematics – Making invisible visible.
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Procedures were main focus in mathematics for thousand years. There were a good reason why is that was math education about. Today math teachers have more difficult assignment than ever.
Today is shortage of mathematician who are able to solve the real world problems.
It is really hard to mathematical teachers today.
**** Exceptionality
I will give you example of sport team using this method won championship.
You measure 200 things and practicly everybody (92,84%) on the planet is exceptional.
Our intuition about geometry a hopeless. What this means. If you have a solid box full of staff most of the staff is in the middle. With three dimensional solid box most of the staff is in the middle. With 200 dimensional box almost everything is outside and the midle is empty. The middle of solid box is empty. Solid but it is almost empty.
Oakland As 2002, 2003 baseball team coach Bill Beane hired mathematician Paul DePodesta. He build a team where no one was exceptional. These were players who all other team ignore. He found players who exceptional in this range of characteristics. Within one year they won national competition. In the movie Brad Pitt is coach.
This works. This means that no one group of people, country need to be at bottom. You can use relatively simple mathematics to identify suite of parameters in wich are people you have in front of you are exceptional. And find the way to use those abilities and fit them into the world. That is hopeful. It is also tell us how our society is structured. We are measuring exceptionality according to a very narrow definition. But in today’s world in particularly being exceptional in range of things is much more important than being exceptional in one thing. Cause if you need someone exceptional in one thing you can hire consultant. This is mathematical result. So it is true.
When precision of mathematics meets the mesines od real world
Departman za matematiku Novi Sad
Why people do the mathematics? Why do humans do mathematics? Well why did people like may become mathematicians? We did so because we found it was fun and it is the challenge. It was always challenging. And we just like doing it. It is like climbing mountains or skiing or Anything you want to do you want to do well. It is a huge challenge to met the challenge.
Why do people payed to do it? Somehow I do it anywhere. I like doing it. Many people around the world are paid to do mathematics or to teach mathematics. So for the society point of view why do mathematicians get paid.
Well because mathematics has always been developed to do things in the world but it’s always been useful and continue to be useful. Because the thing we use mathematics for is to help understand the way to do things in the world. It is one of many ways to understand the world we live in. To the degree that we can do things in the world that we live in. And that’s been the case since going back to Archimedes, Newton, Leibnic, Pascal.
But how thas this work. How does mathematics help us understanding the world. The man trying take this thing called mathematics which is developed over the millennia and specifically over several hundred years to solve problems in the world. And it reached a stage of incredible precision where there was a large amount of mathematics. Actually mathematicians and people in society lost sight to some extend of what it was about. But because we are have problems that we can’t solve without mathematics we have to rethink how we use mathematics to solve problems need to be solved that cannot to be solved with the way we were doing mathematics before.
So let’s just see historically and procedurally how mathematics plays out in the world. You start with some real world situation something going on in the world. For example supposing you want to build a house or you wife want to build a house. You have some land to build a house so you you talk to your family.
Then you find someone who can build it for you. The first person you talk to is a architect. The architect will talk with you over several weeks find out what you want your house to look like, how you want to live in it, what things you want in the house, how big it is, how expensive, how much money you have.
And that architect will take your own description in human terms. The architect takes your description and converts it into a whole list of things that he or she will have to do to build a house. It’s the same that you have said except in a different language. It’s in the technical language of architecture which is a mixture of so bits of mathematics and trigonometry but also a lot of social science, community science stuff you know as well.
Then someone draw blueprints. Actualy trigonometry drawings with angles and measurement, specifications, charts, some spreadsheets next to. It turn it out into a mathematical description. And the blueprint is a mathematical description: the diagrams, the spreadsheets, the costings, the amount of timber, the amount of cement all of that stuff gets written down and calculated.
So now it’s gone from a human description to a technical description in terms of architecture to a technical description in terms of mathematics. So your houses now become a mathematical specification.
Your house has now become and some large matrix of numbers which are going to be followed. But it doesn’t looks like house anymore. It now looks like mathematics. And if the all was the well what you get is what you wanted.
And most people have no idea what was on the right. They don’t need to know. They just need to know that it works. That is how mathematics works all the time in every application.
The part on the right we use call it mathematical model. When you took something in the world that you were expressing in mathematical language, and you start doing calculations, derivations with it your working in a mathematical model of the world. And mathematical models are usefull. Because it feedbacks to the world. This is why society wants people to do mathematics.Society don’t care what happens inside the box. Only the mathematicians care about what happens in the box. What society wants to know is it does the box do what that diagrams say it does. That’s triangle captures the way mathematics fits into the world we live.
Numbers
I give an another example. Few years ago I worked a successful crime series on American television called Numbers. This is about how you can use mathematics to solve crime.
The idea of Numbers is that young, clever mathematician to help his F.B.I. brother solving the crime.The F.B.I. agents would get stumped, would find it difficult to solve crimes, would talk to his brother and the brother would coming and apply some mathematics to solve the crime. Every single episode follow this plot.
When series first appear the critics said this is not believable because the F.B.I. doesn’t use mathematics and that you can’t use mathematics to solve crime. They were absolutely wrong. The first few episodes were based on real cases. The F.B.I. using mathematics for a long time.
So you don’t have to do things like building, which is clearly engineering, building a house full of engineer so you expect there is mathematics. But you could apply mathematics to people.
The F.B.I. agent comes in and checks the crime scene and translates in terms of police work.
Then he gets stuck off. He brings in his brother to do some mathematics. The brother translates the problem from police language to mathematical language. He then does some math. After the first few episodes they invented cases. But the mathematics was always correct.
The very first episode was about a serial killer. It was based on a real case, the real guy Kim Rosmo that used real mathematics to capture serial killer. This is the actual equation what Kim Rosmo invent.He wrote equation that allows you to identify where serial criminal most likely lives. You need 5-6 cases to get enough data.
You can apply this techniques of mathematics to people.
But here’s where the mathematics works and this goes back to Newton-Leibniz inventing calculus, to Kepler looking at the planetary motions, to Archimedes looking a various problems way back then. This is how mathematics always were developed. You identify some patterns in the world. Mathematics is science of patterns. It could be pattern in repetition in a serial criminal, it could be the patterns of the motion of the planets, it could be the pattern of traffic on the highway, it could be the patterns of sales on Amazon, it could be the pattern of the package ruting on Fed-Ex. All of these patterns you can capture them using mathematics.
If they’re regular patterns, if they’re patterns of repeats you can use mathematics on them. You almost certainly needs to use a notation. Or very often you have to invent a new notation it really will suit it’s up that pattern. You then use the notation, take observations, you do some experiments, when you translate that the answers and the observations in some language so you can study more, then if you’re doing the whole thing you can start writing down assumptions or axioms. One of the first examples of this is geometry. You can look at patterns in the world and when you are apbstract them into a formal language you get the language of geometry.
Lines, triangles, circles etc. Then you can write axioms. What it is you are asuming. What are the basic principles of this things are following. And what you’ve done that you can actually start making some conclusion and you can get new knowledge about those patterns by using logical reasoning. And then you can actually start writing down methods, algiritms, equations, formulas so that other people don’t have to go through the process. They can just say my problem is write that problem over there. The people who looks at that problem develop this notation, they develop these conclusions, they develop these formulas. I can use those formulas in this new domain.
So now you’ve got something that other people can use. And if it proves successful it becomes part of mathematics. Newon and Leibniz invented calculus to make precise predictions about motions of planets in fact continuous motion in general. Once they done that generation after generation of university students in mathematics can learn calculus. Without having to go through all of that process. You just learn the rules, match the notation and you can apply to many domains.
So it’s really a competitive technology, the calculus is competitive technology. You can take it off the shelf, you can apply to a problem and they provide us answers. Very very efficient process. Just applied to the world. You now on position where you can generating new cycles you can keep generating new triangles that take you around.
This patch in the middle. That’s the specialist of that sort of in the mathematical models if you like. That is how we have axiomss. You have proves, you have procedures. That what we call mathematics. That’s what mathematics is like in mathematics department. The only people actually in the world who think of my mathematics in that way.
Universities get the whole thing. They talk about axioms, proves, procedures.
About ten years ago the moment Steve Wolfram released WolframAlpha the world of mathematics changed. Because WolframAlpha which you can access free from your smartphone. It’s in the cloud. You can pay for better version but it is in the cloud. That software system all done for you, all the procedures whatever. If you put Wolfram Alpha into your university exam it will get you a big plus. It’s a software system so it doesn’t always get you an A, especially if professor gives you a trick question.
Which means it’s now possible for people to use the whole cycle. Even if they don’t know how to do details of the mathematics. And the reason that is important is that means people whose expertise is not it mathematics but an application domain can use this process. Criminologist, sociologist, psychologist whatever all the experts could use this process on work their way through. And one they have to do some actual mathematics some formulas, some procedures it’s all available for free in the cloud.
That means the mathematics has just changed in the way to be done on the way is being done. Beyond WolframAlpha there’s a whole range of others tools are available. So we now found machines to do all of the school and university procedures of mathematics.
So that’s change a lot. But let me give you an example. For the period about 1985 to 1990 one of the main researche question was what is the information. That became important because computers have been networked. When Cisco Systems was created within six months it have the highest market capital of any company in the world. So this was an enormous industry and yet there was no science of information. It was happening but we didn’t have a science to understand it.
Information
And asking data is always dangerous. You need to understand what you doing. Otherwise you’ll end up in trouble. You might end up in the building things colled social networks which tend to be very powerful weapons when they are weaponized. Which meant the world was building information technologies and have no good definition of information.
Let see what people meant by information. But it turns out to the fourteenth century the word information that’s what it first appeared. It was actually about being smart, beinformed, be educated. Information as it was used was just to describe someone who used of.
Mid nineteenth century change because in the mid nineteenth century you get newspapers. Newspapers turned information forward description about people knowing things to talking about substances.
So information was a commodity of a certain size, weight, shape, form. It came into your house. And then you became informed.
So there was no longer referred to what was in the head of infomed persons. So now we have information as a commodity. That’s a very different notion.
Changed again in the mid twentieth century because when people were starting to connect computers together. These new technologies were simply taking newspapers and putting them online.
But the moment that change happened you were doing something different. Because once you start doing that it’s no longer a quantity that you can measure etc etc
Information used to been something in someone’s head. So historically it’s about going on things going in your head.
So let’s call it something else. Let’s call that knowledge. We also have something called data. That’s the stuff that’s been shipped in the wires or in the newspapers.
Because this new notion of information isn’t in head and it’s not on the newspapers or in the wires. Somewhere between. So information lives between these two domains.
It’s something new. It didn’t really exist in human society before. Implicitly it did. But there was no need to be explicite.
How this thins are linked together? If you have data, if you add meanings to it, if you could interpreted it inside your head then you get information. If you consider the information and make it a variable to guide your activities we called it knowledge. And these are separate domains. The data is the same for everybody. Information derived from the data can vary from people to people. And the knowledge is something higher. Because information is in the middle. It’s a social construct. Knowledge is in the brain. Data is in the physical world.
The spies exchanging information. People don’t. They ask questions, they argue. You only talk about people actually exchange information if you want to draw attention to the fact that something weird is going on.
Serendipity (I will write about this)
We get information because we recognize the certain context. Ringing bells could be all kinds of things. Within our home with the door bells you know it means someone on the door. You can’t see the person you just hear ring the bell.
Dark clouds means it’s like it’s rain. Door bell rings means that somebody at the door. The traffic lights are red means you have to stop.
And then you can conclude something happen or something not happen.
The one I did with Duska Rosenberg about air trafic. We studz communications patterns. What we did was when we took our framework and climb to her data we were able to mine down a set of rules.
They could pinpoint where the criminal was. We could pinpoint where the problem was in the information flow. The way I like to describe it is this : Information in the twenty first century is an ocean and you can just go beneath and no one will knows. You need some stepping stones. Mathematics provides the stepping stones. It’s not a bridge. It doesn’t get you into the answer. But it allows you to make precise statements often enough that you don’t sink in the ocean. Enough huge. Because without mathematics you sink into the ocean. It is too much it.
Again this is just to give you a flavor. This is it. The is available you can get it from Amazon.
It was about engineers, the engineers records, understanding how engineers communicated, what was happening was when engineers repaired the systems, kept records and they were passed from one enginer to the next, then past to the company.
It turn out that things were going wrong because people miscommunicated this documents. That was kind of weird because the documents only have some very specific items of information. There were phone information that was about what the part number etc.
And company couldn’t understand why it’s didnt work because all of the information is in these documents. And yet all of these different parts of the company we’re looking at the documents. They were reading the documents in different ways.
Our task was to find out why this is of how this could improve. We actually spent a lot of time not looking at the documents but looking at air traffic controle.
After several months it turned out that there was no one kind of document even though it looked the said. There were three categories of documents. Suddenly ocean of documents was split into three categories of documents. We have a structure. We have a way to talk about it, way to descibe it, way to be precise about it, and we were able to make conclusions and said : this is what you need to do to fix your problem. And the company did it.
I want to finish that but I’ll just mention the if you’re interested looking about on the book on the left is a technical detail. Or one of them right is a more technical.
QED
STEP in Math