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Jan. 15, 2021 - Epoch Times
33:55
How to Solve Your Problems Like a Mathematician | World-Class Mathematician Daniele Struppa
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And this is a very crucial time.
You need to be willing to entertain the possibility that you are wrong.
You need to see colleagues that will try to poke holes into your arguments.
If you don't do that, you will fall victim to some kind of a confirmation bias that happens in every discipline.
So you get convinced that this is true and you are unwilling to look at all the evidence that may show you that you are wrong.
So as we develop an idea about whatever mathematical object it is, we need to be able to think that maybe that idea is not the right one.
And so I think that all of those techniques, so to speak, are very valuable when then you try to apply to other problems.
Welcome to California Insider.
My guest today is Dr.
Daniele Strupa.
He's a mathematician and president of Chapman University.
He received his doctorate in mathematics from the University of Maryland and has co-authored and edited more than 10 books.
Today he's going to share with us how we can use critical thinking to conquer the ideological divide in society.
And how math can also be used in our day-to-day lives to help us solve complex problems.
Daniele, welcome.
Thank you.
So it's great to have you on.
We want to talk to you about a lot of things today.
You have a way of using math for problem solving.
And I want to talk to you, before you get into showing that to our audience, I want to talk to you about this big divide that's going on in our country.
Right now, the left and right are very divided.
And I sometimes think to myself, what if these people come at each other with guns?
It's getting to that point.
Can you tell us what's going on from your perspective that's driving this divide?
Well, I have to preface my comments by saying I'm not a political scientist.
And that's probably a good question for a political scientist.
But I think as a citizen, I'm very worried.
I don't like what I see.
I think there is a breakdown in communication between the two sides.
And I think that we have become so divided to the point that we can't see, we can't admit that our opponents may have some good ideas.
And that to me...
It's connected with what I think is an issue with the ability of people to practice critical thinking.
So you and I may have a different idea, but when we sit together to discuss a topic, I need to have enough goodwill in my mind To know that even though I think I'm right, obviously, that there is a possibility that you could be right.
And when I listen to you, I need to listen really being willing to change my mind if necessary.
And I would expect the same from you.
Of course you sit and you and I have a different position.
You think you're right and that's totally fine, but you need to allow me in your head the ability or the possibility that I could be right.
And this is really, ultimately, is connected with our ability to do what is called critical thinking, which is the ability to control the way you think in a way that analyzes the various aspects.
If I sit down and I already know everything, then I automatically exclude that possibility.
And I think that our society, for some strange reason, is being driven in that way.
We hear this from the media.
I have to say, I watch TV and I watch both sides of the aisle and I've come to the point in which I don't really trust either one of them.
And I see again this lack of goodwill and the ability to say, okay, let's look at the problem and let's see whether my opponent may have actually some ideas that might be helpful.
I don't think there is one depository for truth, which is whether the right or the left.
And I'm disappointed.
And I think that A big component of this is the work that schools need to do.
And what is that?
Well, you know, schools have to teach people the ability to look at the problem and understand the problems are usually more complex than we think they are, and they require kind of a Multifaceted perspective.
You know, somebody asked me, how do you know that the university is training people to be better citizens?
Now, we don't have specific tests for this, but I have something in my mind that I always thought would make sense.
And the idea would be, you ask a high school kid how he would solve a very complex problem.
Let's say poverty, okay?
So how come that we are a very rich country and there is so much poverty?
Now, a young kid most likely would come up with a very simplistic solution that reflects probably what he or she lives in the home.
So depending whether they're more on the liberal side or the conservative side, they would have different solutions, but equally simplistic.
The university would do a good job if after four years you ask the same question.
And the student would probably answer, I don't know.
And that seems contradictory.
What do you mean?
All the learning is about you don't know?
Well, the answer I don't know is the right answer because poverty is a very complex issue.
So what I would expect the student is to say, well, you know, I don't know.
I don't have a solution.
But I think that the aspect that we need to look at are the following.
And then I will hear a conversation about demographics.
I will hear a conversation about the political system.
I will hear a conversation about the sociology of wealth transfer.
And so, little by little, The student that has really learned how to think would portray for me a really complex situation.
A situation that probably will not allow any single individual to find a solution, but would require a multidisciplinary team.
You want to have the economy.
An economist by itself cannot solve the problem of poverty.
A sociologist by herself cannot solve the problem of poverty.
What do you need is really a complex group of people.
So to me, that is what universities, but also high schools should start teaching more.
We need to convince people that if you have a quick answer about the problem, most likely you haven't understood the problem at all.
And now, how does this connect to math and the problem-solving way and math?
Well, that's a good question.
I mean, math is the problem-solving discipline By itself.
I mean, all we do as mathematicians is to solve problems.
And we like problems that are difficult because they're the ones that kind of push our, that challenge our intelligence.
And so Even though the complex issues like, say, poverty that I described before cannot be addressed simply with a mathematical formula, what we learn as mathematicians, the discipline that we learn as a mathematician, the mental discipline, is very helpful.
I think it can be translated.
So the first thing that a mathematician does when he's posed the problem is to try to define the objects he's working with.
So you want to understand what you're talking about.
And too often I hear people talk about stuff when they clearly don't know what they're talking about.
One of the big discussions in this country is about what is fair.
But you don't really hear a deep analysis of what the word fair means.
So as mathematicians, we are trained that we need to figure out, first of all, what are we talking about?
We collect data.
We do a lot of examples.
So if I'm trying to figure out a strange property of numbers, I'm going to run a lot of examples to get a sense, to see whether I can identify a pattern.
Is it something that always occurs?
I think that that's very important.
So data needs to drive much of our understanding.
A mathematician, after a while, will probably start thinking that he or she has an idea about what the solution to the problem is.
Partly because of your experience, partly because of the examples you run.
You get to a point where you say, okay, I think I know what's happening here.
And this is a very crucial time.
You need to be willing to entertain the possibility that you are wrong.
You need to see colleagues that will try to poke holes into your arguments.
If you don't do that, you will fall victim to some kind of a confirmation bias.
That it happens in every discipline.
So you get convinced that this is true, and you're unwilling to look at all the evidence that may show you that you're wrong.
So as we develop an idea about whatever mathematical object it is, we need to be able to think that maybe that idea is not the right one.
And so I think that all of those techniques, so to speak, are very valuable when then you try to apply to other problems.
Can you show us some of those techniques, how you would solve the problem with using math?
Well, there is one very famous example.
And the example is you draw a little square with nine dots.
Three dots, three dots, three dots.
And then you ask the participant to find a way with the pencil to cover all the dots with only four segments without ever lifting the pen.
And then if you try to do it, the majority of people get stuck, because maybe you start with the first three dots, and then you go horizontal, the second three, then you go down, and then you can't finish it, because if you finish the one in the middle, it's going to be empty.
So then you say, okay, well maybe I'm going to try a diagonal.
So you do diagonal, and then vertical, and diagonal, and then now you're leaving a point on the side.
So you're going to be struggling with this.
What you are doing here is that you are adding to my problem a condition that I never mentioned.
You add the condition that you shouldn't go outside of the square.
I see.
The solution to the problem is, in fact, you start with a segment that covers three points and goes up one more step.
And then you can show that you can do the little drawing.
So this example is useful because it's telling you, it's telling us something about us, what we are as human beings.
We have the tendency of adding conditions that are not really there.
And I think that that's one of the things that we always need to look at when we try to solve a problem.
Just a couple of hours ago, I was on Zoom with a group of governors of the university, and we are looking at how we redesigned the entire approach to mental health for our students.
And it is clear that in the past, we also did the same thing as the example of the nine points.
We were blocked by thinking that, for example, we should be solving the problem within the institution.
And we shouldn't instead consider outsourcing methods that can be helpful to help the mental health of our students.
So the idea that you add an extra constraint is very common in life.
Without really knowing that you're adding it.
Exactly.
You don't realize that.
That's a danger.
It's kind of a version of confirmation bias, but it's even more insidious.
Because in confirmation bias, you have an idea, you stick to it.
Here you actually add a constraint to your problem.
A constraint that is not there.
And once you remove the constraint, You realize that there are other possible solutions.
That's why the sentence, thinking out of the box, that's what it means.
That you are removing the constraint, you are inside the box, and that doesn't allow you to solve the problem.
Another typical thing that is very important in mathematics, and it's applicable in every situation, is what they call the functionality fixedness.
So you think that an object is only good for what you always thought it was good.
There is a famous, simple joke or story.
A guy is given a barometer and he has to use the barometer to see how tall the skyscraper is.
So how tall is the Twin Towers?
You would have said tomorrow it's 9-11.
So the typical answer is you look at the barometric pressure here, you go to the top of the twin tower, you look at the barometric pressure, and then that tells you you can transform the difference in barometric pressure into altitude.
And that's a standard answer.
And then the next step of the game is, okay, suppose the barometer is not working.
How do you use it to measure the height of the skyscraper?
That requires you to look at the barometer, not as a barometer, but as something else.
And so the answer, there are two standard answers.
One is that you trade the barometer for somebody that is going to actually do the measuring.
But usually the answer that we hear is that you still go to the top and then you drop the barometer and you time how long it takes to hit the ground.
That's interesting, yeah.
So now you're using another property.
So the barometer is not a barometer anymore.
It's a rigid object, and we know very well what the law is that tells you what the velocity is.
So you can easily, once you clock the time, you can easily calculate the ion.
So what is the point here?
The point is that you are using...
An object for something that wasn't used to be useful.
In mathematics, that's how most of the great results in mathematics are obtained, is when you are able to show a connection between two areas that is completely unforeseen.
Maybe the most important example, well, the most famous example, is It's something that a mathematician would know and I don't expect anybody else to know about it.
It's called the Riemann's conjecture.
So Riemann was one of the great mathematicians of two centuries ago, one of the giants, and he was interested in figuring out How many prime numbers there are?
Well, we know there are infinitely many prime numbers, but how often do they appear?
At the beginning, there are lots of them, right?
2, and 3, and 5, and 7, and 11, and 13.
So there is a bunch of them.
But as you move farther away, they get less and less.
So his question was...
How many of these, I mean, how frequent are they?
Can I figure that out?
And the answer to this, this is a question about integers, about numbers.
It turns out that the answer to this becomes a problem that has to do with integration, that has to do with complex numbers, complex theory.
So the idea is that you need to be able to use something that is completely different from what it seems that you're looking at.
And I think that that's true in every significant problem.
We need to have the ability of Thinking outside the box, which often means using instruments that are not originally designed for that purpose.
So math helps with that?
Math helps in creating the frame of mind.
So I'm not saying...
I want to make sure that I don't mislead our viewers.
I'm not saying that math gets applied to all these problems.
There are also problems that math can't really solve.
But the frame of mind and the mental discipline...
That type of thinking.
Exactly.
That type of thinking is what is helpful.
So first you're willing to be wrong and you're willing to look at...
You want to actually look at the facts.
You want to look at the solid facts that are involved in a situation.
And then second thing is you're willing to think.
It will train you to think outside of the box.
That's right.
And you need to be willing to set aside what...
Your original instinct may be.
There are lots of problems in math that I look at, and I think I know what's going on.
But I need to be wise enough to know that that may not be the case.
So I fight against my own confirmation bias.
That's one of the problems where you were saying before about the divide, the social divide.
The reason why the divide is not breaking down is because people are affected by this confirmation bias.
So now you have two political groups.
Something happens.
Every group is going to draw from that event only the portion of the event that confirms that they're right.
And that's why it's almost impossible to simply demonstrate why one group or the other may be wrong.
It's impossible because even facts It can be deconstructed.
The fact is the same, but people will draw from the fact different conclusions.
They will take out of a certain complex event only that part of the story that confirms that they are right.
And that is a significant problem.
And I think that it's not just mathematicians, but generally speaking, scientists are trained to fight against that.
And do you think we're missing this type of critical thinking in the society now, in the education system or media in general?
We haven't been educating the children to, or this generation of people?
I don't think we're doing a very good job.
And I've been critical of our educational system for a long time.
I don't think we take enough time to build that kind of mental discipline that is necessary.
And I love this country, so this is really not designed to be a criticism of this country, but America is not exactly an intellectually driven country.
In this sense, it's very different from Europe.
And sometimes I feel that we don't understand the importance that serious Hard work has in the training of young people's minds.
You know, in Europe, for example, people study things that now in this country almost disappear.
People study Latin, people study Greek, and the question is why would you want to study Latin and Greek?
These are not languages that anybody speaks and you're never going to use them.
You're not going to use it in your work.
You're not going to use it anywhere.
But what they do is that they're complicated languages to learn, and they actually force you to a constant exercise of critical thinking.
Because when you translate a sentence, say from English into Latin, you have to analyze the sentence in a much deeper way than only to translate between other languages.
This exercise forces you to look in a deeper way to the structure of the sentence.
That exercise is irrelevant from the point of view of learning new languages, but it's going to be very important because it's an exercise that remains part of your frame of mind.
And I think that sometimes we are going in a direction, I'm not suggesting that people should at all cost study Greek or Latin, but I'm offering this as an example of a very rigorous And I think that we have been watering down the rigor in schools.
Our schools are driven by, I think, the wrong metric.
They're driven by the metric of graduation.
The better school graduates more students.
Well, yeah, it's easy to graduate more students, but are you actually teaching them?
Are they actually learning?
Are they actually growing?
That should be really what we measure upon.
And I think that in this tendency, this desire to improve our graduation rates, generally speaking, education has been lowering the level of effort the students need to make.
And it's that effort that trains people.
Quite frankly, when we teach math to students, Most of the math we teach is irrelevant.
I can look at you, I can ask you, do you know how to divide two polynomials?
It doesn't matter if you know how to divide two polynomials.
Nobody ever divides polynomials.
I do it because I'm a mathematician, but nobody else ever divides polynomials.
So the question is, why are you teaching that stupid stuff?
Well, we are teaching because it's a rigorous process.
It's a process that doesn't allow you to pretend that you know.
That doesn't allow you an easy way out.
It makes you...
It forces you to think in a rigorous...
And that to me is what we have been lacking in our system.
So that type of thinking, that type of mindset is missing now.
I think so, and it worries me.
Again, I'm not advocating Greek, I'm not advocating math, I'm not advocating any specific discipline, but I think whatever we teach them, we need to teach in a way that is sufficiently rigorous that will make the students sweat at their desk.
Do you think that's been replaced by other things that are more emotional, ideological, or is it...
Certainly, it seems to me that there is a great...
I mean, we see that people make emotional decision or emotional judgment on complex issues.
That sometimes cloud their judgment.
And I think that that's, you know, maybe a consequence of our lack of rigor in the school system.
I can't really draw a connection between one another.
I have to be careful myself not to say something that is not accurate.
But I think that there may be somewhat of a connection there.
I have to, you know, I quote...
For you, and I can't quote by heart because I don't remember this, but there is a guy that I have great admiration for, and you might be surprised by the person I'm going to mention.
His name is Antonio Gramsci, and he was one of the founders of the Italian Communist Party.
And he's one of the most Interesting and intelligent people that I think my country has ever had.
He went to jail for many years during the fascist regime.
And during that period, he wrote a huge amount of work.
And this work is called the Prison Notebooks.
He died eventually in jail.
And he was a true intellectual.
He writes a really interesting paragraph about the kind of school that we need.
And he makes the same point that I just made before.
He makes the point that by lowering the quality of the school...
Now, he's looking at this from the point of view of the proletariat, okay?
So he's looking from the point of view of the downtrodden, the lower class.
How do we help them?
And his point, I wish people were listening to today, especially people who care or claim to care for the...
He makes the point that a school needs to be rigorous, because you need to give everybody an opportunity to become a well-developed intellectual and a well-developed citizen.
And he makes the case, actually, he talks about Greek and Latin, and so here we are looking at somebody that clearly comes from the very extreme left side of the aisle.
The founder of the Italian Communist Party.
But he was a man of great intellectual rigor.
And so he was arguing that the schools have that role.
And if you think about it, this is true even now.
The best way to help the socioeconomic divide is to provide everybody with really good education.
And if we don't, the people who are going to suffer more are the people who come from a lower socioeconomic level.
Because the people whose dad is a lawyer, well, it doesn't really matter.
They're going to do well no matter what.
The people whose dad is a medical doctor, they're going to do well.
Are the people who don't have the family structure behind to help them succeed, then we have a stronger responsibility.
But the stronger responsibility cannot be satisfied by Be nice by just letting them go through.
It has to be satisfied by strengthening their results, by providing the intellectual discipline.
And it saddens me that I think sometimes we forget that, and we choose the wrong path.
Towards the growth.
Now, what do you recommend for people that are watching?
How can they use this concept that you're talking about with math to apply that into their daily lives and to their career and to their future?
So you're asking how can people use these ideas in their everyday life?
The way I would answer this question It's by, first of all, asking people to make a real genuine effort to examine their own bias and preconceived ideas.
We all have them.
So there is nothing wrong with being biased, you know.
It's very natural.
I have some ideas, and of course I think I'm right.
Otherwise I wouldn't hold them.
But I think that it's important for people to make that effort.
So, for example, one of the things that I always recommend my students, watch the TV channel that you despise.
Don't watch only the one you like.
So, for example, I'm more on the conservative side.
But I watch a lot of more liberal media.
It sometimes upsets me, but it's a good exercise because I watch it with sufficiently open mind that sometimes I do say, hmm, actually that's a good point.
I wonder how my guys would answer that.
That step of being able to accept an opposing viewpoint, And actually looking at it as an interesting intellectual challenge, it's very important.
If we all were to practice this a little more, I think we would be more open to different ideas.
I want my liberal friends to watch Fox News and every once in a while, not always, but every once in a while to say, well, I hate this guy, but this guy has a good point.
Just like I do when I watch MSNBC or CNN. It doesn't have to be always, but it injects in us the doubt that in fact we are not the owners of truth.
It's not 100% one way or another.
It's not.
And everybody can bring good ideas.
So my advice for, you said, how you apply this approach is really make this effort.
Make this effort to sit with people that think differently from you.
We have this danger of this confirmation bias, as I mentioned before, which is really one of the great enemies to critical thinking.
The confirmation bias and this eco-chamber that we create denies us the ability to challenge our own views.
And I think it's very important that we do that in every area of our life.
And I'm going to give an example that...
People sometimes find strange, you know, I'm a Christian, I'm a Catholic, I believe in God, and yet I question that faith all the time.
My faith is asking me to believe things that are really, really hard to believe.
I have to believe, as a Catholic, that Jesus resurrected after three days.
That is a huge request.
I'm a scientist.
So now I'm faced with a request on my brain that is so strong.
But as I like to say, it is this ability to doubt my own faith that I think makes me a better Christian.
I read about it.
I talk to other people.
I talk to people both of faith and not faith.
And that, to me, has been very important in my path as a person of faith.
I like to say, there was once a beautiful commercial, I can't remember who was using it, but I used to climb big mountains, and I like this commercial.
It said, without fear, there is no courage.
I like it because it tells you that, what does it mean to be brave?
Well, to be brave means that you know that there is a danger.
You're scared of the danger, and you win that fear.
If you don't understand there is danger, you're just, you know...
Big risk, yeah.
You're just silly.
You don't understand that you could be dying doing a certain thing.
And I think that the same is true for faith, I like to say, and without doubt, there is no faith.
And of course, there are lots of people who have great faith and no doubt, so I'm not being absolutist here.
But for me...
My faith is strengthened by the doubt.
And this is part of what I'm saying.
In other words, it's the ability which you have to cultivate on a daily basis of looking back at what you believe and say, well, I could be wrong.
So, even if you're not good at math and you went to school and you didn't apply the math concepts, you can still apply this type of thinking.
I think so.
You have to question your own judgements, question your own knowledge.
What math does, but you don't need to have taken math for that, is it builds kind of an ethical discipline.
And that's what I think people need to do, is an ethical discipline.
Which is the ability of always question what you think in a real way, though, in a genuine way.
Because it's very easy to say, well, I'm open-minded.
But we are not very much.
It's an effort.
And I know because it's an effort for me.
I sit with people...
And I know when I see that I think they're really wrong.
And so then I call myself on that.
I say, ah, you can't do that.
This violates your principle.
You have to give the person a chance to really convince you.
And it's a very difficult thing and I struggle with it.
I don't want the viewers to think that I'm preaching from the top because I do all of this.
No.
Too many times I fail.
But I think it's an important form of discipline.
And yes, you don't need to be a good mathematician to do that.
To me, it was taught by math.
When I started being a mathematician, I realized how important that was if I wanted to be successful.
You know, a few months ago, I wrote a paper, I discovered a theorem, which I thought was interesting, but I thought it was too good to be true.
And so I applied this principle, and I called two friends of mine from Italy, who are experts in that discipline, and asked them, Do you think I'm right?
I don't know.
And they said, no, no, you must be wrong.
But they couldn't show me where I was wrong.
So then I called two more friends of mine from Israel, also experts.
And they also said to me, I remember we're doing a Zoom, and they said, oh, no, Daniele, you must be wrong.
Let us work on it and we'll show you where you're wrong.
And then a couple of days they called me and said, not only you are right, but now we understand you're resulting in even a larger picture.
So we wrote a very nice paper.
The point of the story is not about the theorem.
The point of the story is that I was looking for people to disprove My viewpoint.
I wasn't looking for people to tell me, good job, this is a, oh, that's a great, no.
I was trying to figure out if it was a mistake.
That attitude in mathematics is necessary.
You can't be a good mathematician unless you are willing to constantly entertain the possibility that you'd be wrong, and in fact you want your colleagues to double check what you're doing.
You're open and you're willing to accept.
Math is our daily discipline, so it doesn't take anything special.
What I'm suggesting here in this conversation is that you have to take that self-discipline and move it to every other area of knowledge.
And further, to your question, I'm suggesting you don't need to have gone through the math to go there.
You can just forget about the math, you don't have to divide the polynomials, and go straight into the practice.
The practice being?
Always assume that you may be wrong, even though you think you're not.
And that is a difficult exercise.
Listen, it's difficult even between husband and wife.
You mentioned to me this big political divide, but people experience this challenge at home, husband and wife.
Can you have an argument with your wife, not you personally, you generally, have an argument with your wife and have in your mind the idea that she might be right?
I challenge everyone of my viewers to think about this now.
It's very hard.
When you have an argument with your wife or with your husband, this is gender neutral, It's very hard to sit at the table and think that your husband is right or that your wife is right.
So can you imagine how difficult it is then?
These are people you love.
So then the problem becomes national.
So I think that that's kind of the discipline that we all need to do, is to be really willing to accept that.
And I think that it's the first step towards a more critical analysis of what we believe.
Well, is there anything else you want to touch on?
Well, I could talk about the prime number theorem of Riemann, but that would be the immediate...
It would be way over my head.
I'm going to turn off the television right now.
So, no, I think that this is...
I appreciate you asking this question.
I'm passionate about the role that education can play in a society.
Gramsci, again, I'm going to go back to one more time to my former communist hero.
Gramsci said something also very deep.
He said, every society...
needs people that are willing to engage in this very painful and difficult, disciplined work.
And I think we cannot forget that.
Though we may not ascribe to communist ideas, These fundamental ethical principles are very important.
This is a man who spent most of his life in prison, and he was spending every single day working for 10 hours a day on this work.
And I think that that's one of the lessons that I take.
The reason why we still remember his name, we still read his books, even people like me, who is not a communist, and so I disagree with some of his conclusions, but we have this tremendous respect For the strength of the intellectual discipline.
And that's what I think we all need to remember.
So I'll leave it at that, and I thank you for stimulating me with these thoughts.
Thank you so much.
It's great to have you on.
Thank you.
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