Celebratio Mathematica

In­grid Daubech­ies is the first fe­male full pro­fess­or of math­em­at­ics at Prin­ceton Uni­versity. She re­ceived ten­ure there in 1993 after work­ing at Bell Labs. She was born and raised in Bel­gi­um. For this in­ter­view, I ac­tu­ally traveled to Prin­ceton and met with her in her of­fice at Fine Hall. The full tran­script of the in­ter­view will be re­vealed in parts.

Ken: Thank you so much for agree­ing to sit down with me for this in­ter­view! I think that the nature of the math­em­at­ics pro­fes­sion re­mains a bit elu­sive to our mem­bers so the op­por­tun­ity to speak with one of today’s premi­er math­em­aticians is a very for­tu­nate event for us. I’d like to start by ask­ing you about math­em­at­ics gen­er­ally…what is it to you?

Prof. Daubech­ies: I’ve done work that is pure and work that is much more ap­plied. I did this work on wave­lets — it doesn’t really mat­ter what they are for this pur­pose — but the thing is, it is something that is mo­tiv­ated by con­crete ap­plic­a­tions in sig­nal ana­lys­is, in par­tic­u­lar, for im­ages and it’s use­ful for those and it in­volves some spe­cial func­tions.

Now, when people talk about those they al­ways feel, I feel, every­body feels, they have been con­struc­ted. Now, com­ing from a math back­ground, you prob­ably know that many people feel in math­em­at­ics that math­em­at­ics is dis­covered and it is an out­side thing and they dis­cov­er it and they get in­side and so on. But about these things [the spe­cial func­tions] people feel that they are con­struc­ted. But the feel­ing, I can tell you, from work­ing on them and fig­ur­ing it out and so on is ex­actly the same wheth­er you do pure math or that. So, I try to find out why is that?

I think it’s partly be­cause most of math­em­at­ics was mo­tiv­ated by phys­ics. And in phys­ics, you try to de­scribe things that ex­ist while when you do it for tech­no­logy, it is, ob­vi­ously, that you try to find ways, pat­terns, and so on, that help you con­struct things your­self. But the math part of it feels the same and even people think of it dif­fer­ently. So, in the math, where’s the bound­ary [between the dis­covered and the con­struc­ted]? Where do you feel that you go over the bound­ary? And if you start look­ing, you don’t find it. I be­lieve there is no bound­ary. I be­lieve that all math­em­at­ics is something that we con­struct. I think that math­em­at­ics is something that does not ex­ist “out there”.

I think that math­em­at­ics is, in a sense, a very hu­man thing. I mean, it’s our way…it’s the name we give to when we try to really pre­cisely and ar­tic­u­lately de­scribe struc­tures that help us in de­scrib­ing nature and study­ing nature and con­struct­ing things and so on. But that whole pat­tern of pre­cise think­ing and mak­ing things fit and try­ing to find pat­terns, and be­cause we can­not build it all, we have to build smal­ler build­ing blocks that we then use in build­ing big­ger things and that’s why we build all these math­em­at­ic­al con­cepts and the­or­ies. Of course, it is very nice when the thing that you de­veloped here can be used there. To me, that’s math­em­at­ics.

So, I think there are lots of math­em­at­ic­al types of reas­on­ing where people don’t write a single for­mula if they try to really very pre­cisely and un­am­bigu­ously reas­on their way through. I con­sider that a form of math­em­at­ics. In­stead of say­ing math­em­at­ics is this ab­stract thing for which you have to be a kind of weird mutant to be good at it, I say, it’s not true. I mean it’s something that we all share. It’s a very hu­man thing. I could con­ceive of us meet­ing an ali­en race that would do its math com­pletely dif­fer­ently and we would have great dif­fi­culties in un­der­stand­ing them. I hope we would learn, but I’m not really sure we could, but you hope.

So, I don’t be­lieve in this uni­ver­sal thing. I feel it is a very hu­man thing.

So it’s also something at which some people are more gif­ted than oth­ers…just like you have Olympic ath­letes and oth­er people. But, I mean, there’s nobody telling all these jog­gers on the road that they shouldn’t run be­cause they will nev­er be an Olympic gold medal­ist.

There are now all these Sudoku puzzles that are so pop­u­lar. I thought once, “oh, it’s fun…it’s not math.” Well, that’s ri­dicu­lous. It’s not high level math, but it is def­in­itely math and it is pop­u­lar be­cause people like do­ing that kind of fig­ur­ing out. It’s ac­tu­ally much sim­pler than every­day life be­cause you have such pre­cise rules. I mean, you only have a few rules and that’s it. With­in that, you have to fig­ure it out.

Now, wo­men and math…I think it is great to have ven­ues where you are es­pe­cially for girls. Ac­tu­ally, I’ve been think­ing…I’m ori­gin­ally from Bel­gi­um and Bel­gi­um is a coun­try that is big about car­toons. Now, every­body is about car­toons be­cause people like manga and so on, but Bel­gi­um was big on that way be­fore, and they were very dif­fer­ent from Amer­ic­an car­toons, al­though they had star­ted from Amer­ic­an car­toons. It star­ted after the Second World War when Amer­ic­an car­toons were no longer ar­riv­ing. So, for all the series that were on­go­ing, they star­ted writ­ing se­quels. So a whole in­dustry star­ted.

There are a couple of her­oes in Bel­gi­um called Suske and Wiske,1 and Wiske is the girl. But math­em­at­ics, for a com­pletely dif­fer­ent reas­on is called wiskunde. So I was think­ing of ap­proach­ing the es­tate of that car­toon­ist to do something like wiskunde with this girl as the em­blem. Partly be­cause there is, in many coun­tries in Europe now, a short­age of people tak­ing math­em­at­ics — and as a res­ult, a short­age of math-trained high school teach­ers which is dis­astrous. I mean, I can see what it does in this coun­try. It didn’t hap­pen be­fore in Bel­gi­um and in Europe, but it is go­ing to hap­pen. And so, when there is a short­age, all of a sud­den, you might not have any­more this thing of, “it is for boys”…if there’s nobody. And maybe you can then, and this has happened be­fore, that in things that were no longer at­tract­ive to any­body, it was a good niche for wo­men. This is a sad thing, but…def­in­itely, I see it in every pro­gram where you have wo­men that as soon as you have sev­er­al wo­men its fun to­geth­er. Be­cause they are so sel­dom in the sur­round­ings where there are oth­er bright wo­men also in­ter­ested in math.

Ken: Yes, that’s one thing I love about Girls’ Angle. It’s a very dif­fer­ent feel from the co-ed classed I’ve taught. We also be­lieve that when the mo­tiv­a­tion is there, the learn­ing fol­lows nat­ur­ally. So we spend a lot of time mo­tiv­at­ing an in­terest…get­ting the girls in­ter­ested in some math prob­lem. And then, once they’re hooked…just fa­cil­it­ate.

Prof. Daubech­ies: One thing I no­tice when I teach, I try to al­ways re­late things. I have classes where I teach for non-math ma­jors and where I try to re­late to con­crete ex­amples. I put in a lot of ex­amples with my kids and with re­la­tion­al situ­ations and so on and per­son­al an­ec­dotes. And I find that I of­ten re­late much bet­ter with wo­men…I mean that style of teach­ing relates much bet­ter to the wo­men in the class than with the men be­cause the men con­sider it not ser­i­ous. But the wo­men feel, oh yeah, this does make sense.

Ken: Have you thought about modes of teach­ing math­em­at­ics that might ap­peal more to girls?

Prof. Daubech­ies: [paus­ing] I don’t know about that. I’ve read about it, but I don’t really know about it. I think dif­fer­ent groups have dif­fer­ent feel­ings about it. I haven’t done suf­fi­cient ex­per­i­ment­a­tion with boys versus girls in the classroom. I could ima­gine that there might be in­deed dif­fer­ent modes that might work bet­ter. What I do no­tice is that a cer­tain type of be­ing in­form­al helps me make a bridge to the girls while it some­times hinders with the boys. Not al­ways, but it some­times does. With some males I have to do some kind of one-up­man­ship in or­der for them to take me ser­i­ously, which, with girls, they just clam up. I mean, if people do that with me, I clam up too. I mean, I don’t want to work with col­lab­or­at­ors who shout. They can be great math­em­aticians, but they can go shout some­where else. Now, it takes all kinds of per­son­al­it­ies, so none of these are hard and fast rules, I’m sure, but it’s def­in­itely the case that with there be­ing so few wo­men that if on av­er­age, wo­men like a less au­thor­it­at­ive touch and a more col­lab­or­at­ive touch and if they don’t get it that will dis­cour­age them even more…I don’t know if that is something that works…it could well be.

Ken: How did you get in­ter­ested in math­em­at­ics?

Prof. Daubech­ies: Well, I’ve al­ways been in­ter­ested in fig­ur­ing out things. I re­mem­ber when I was little I was al­ways ask­ing ques­tions…and, ac­tu­ally, re­sent­ing it when my fath­er would come with an half hour ex­plan­a­tion in­stead of two sen­tences, like all kids. But I would want to fig­ure out things. For in­stance, I re­mem­ber…I must have been about the age of the girls at Girls’ Angle…I re­mem­ber once be­ing asleep in a car and wak­ing up and I could see the neon street lights through the green strip Spec­tral Lines at the top of the wind­shield. I knew that this green thing dis­col­ors everything, but the neon looked just as or­ange as al­ways. It wasn’t dis­colored. I mean, I moved my head and looked at it through the clear glass and the green glass, and it looked ex­actly the same. And I was really puzz­ling about that. And then later [I real­ized that], well it’s be­cause it is al­most mono­chro­mat­ic. It ac­tu­ally has a double line [see box at right] but it’s al­most one mono­chro­mat­ic line. And, of course, it was dim­mer but what got through had to be the same col­or. A green thing doesn’t shift [the fre­quency of the light]…it may ab­sorb something…but it doesn’t shift. It ab­sorbs some things more than oth­ers, so when you use a com­pos­ite light, you have a dif­fer­ent dis­tri­bu­tion so it may look dif­fer­ent. But when you have a mono­chro­mat­ic light, either it doesn’t come through at all or what comes through has to be that col­or. That really brought home to me very pre­cisely what mono­chro­mat­ic light is. I mean, that’s not math­em­at­ics…that was a phys­ics thing, but still…I no­ticed things and I wondered, how is that pos­sible? How can that be? And so on. And I’ve al­ways had that. But I also had dolls, like many kids and like many girls and I would have a peri­od where I wanted to make clothes for them. And then you ask your­self, how can you, from a flat piece of cloth­ing make something that fol­lows all these curves? And so, that’s when you dis­cov­er, that you have the shap­ing and so on and so you dis­cov­er how to make a non-flat thing out of something flat.

So I al­ways wanted to see how things worked. I al­ways like to put things to­geth­er too. I still like work­ing with my hands a lot and like see­ing things that fit nicely to­geth­er. So, in that sense, I’ve al­ways been in­ter­ested in things. But also, I found it really easy be­cause it was so…I mean, you fol­lowed rules and you would get there.

My fath­er came from a blue col­lar fam­ily. He was the first per­son in the fam­ily who ever went to col­lege. And his par­ents came from a coal min­ing re­gion. So when in high school, the teach­er said you should let this boy go to col­lege, it nev­er oc­curred to his par­ents that there were many things you could do in col­lege. They knew the coal en­gin­eers and they knew doc­tors and those were the people who went to col­lege. And he couldn’t stand blood so he be­came a coal min­ing en­gin­eer. It was very, very lo­gic­al. Later, I think my fath­er would have pre­ferred to be a sci­ent­ist. He had great re­gard for sci­ence and he was try­ing to learn more sci­ence on his own. So he tried to teach me math and es­pe­cially phys­ics. That’s prob­ably why I went in­to phys­ics and not math­em­at­ics at first. So, he would teach me the math so I could do the phys­ics. He taught me cal­cu­lus early on. I thought it was really easy…and it is easy…there are just a couple of rules…if you are will­ing to take it like that…as a game with rules…then it is really easy. It’s do­ing things with it that makes it some­what harder. I mean, when you un­der­stand what you are do­ing, it’s ac­tu­ally not hard. I’m not an ad­voc­ate of teach­ing chil­dren cal­cu­lus at that early age, but I thought there were oth­er things in life that were much harder than that.

Ken: How old were you when you did this cal­cu­lus?

Prof. Daubech­ies: Twelve. But to me, by and large, it was ma­nip­u­lat­ing sym­bols. I don’t think I could have solved a ser­i­ous mul­tivari­able prob­lem. That came later. But one vari­able cal­cu­lus…there’s not much to one vari­able cal­cu­lus really. And the idea of con­tinu­ity and con­ver­gence — it’s a really in­tu­it­ive idea. If you start do­ing the ep­si­lon-delta stuff — that’s an­oth­er thing — that’s to make it pre­cise and ar­tic­u­late, but if you start by say­ing 3, 3.1, 3.14, and so on, it’s ob­vi­ous! I mean, that’s an­oth­er ex­ample I use to show that math is con­struc­ted…it’s ob­vi­ous to us now and it’s ob­vi­ous to kids in ele­ment­ary school, that you have a num­ber line and that you can pin­point things like that and the more decim­als you have, the more pre­cise you are. It wouldn’t have been ob­vi­ous to the Greeks or Ro­mans; they didn’t have that nota­tion. The nota­tion already tells you. It has be­come such a part of what we know that we know it at a very early age. Eu­dox­us had all these prob­lems with this no­tion. He had a glim­mer­ing of it — in fact, I think he really had a feel­ing for it. But if you see all the con­tor­tions you have to go through in or­der to define it really pre­cisely — it is quite a bit of con­struc­tion. But we don’t have that prob­lem any­more be­cause we have nota­tion for it. I mean, con­tinu­ity…you have to have it…it just says we can make sense of the world…that if we didn’t meas­ure things with ab­so­lute pre­ci­sion…if we are al­most there…well, then, the res­ult will be math­em­atician and al­most there as well. That hap­pens in most cases…not al­ways…but it hap­pens in most cases. We’re all used to that. So, all these con­cepts I feel are very nat­ur­al.

Ken: There are some girls who do get the curi­os­ity about some phe­nom­ena they no­tice and they ask them­selves, “why did that hap­pen?” or “how does that work?” but then, an­oth­er thought in­ter­jects it­self, and they think, “oh, it’s prob­ably too hard for me to un­der­stand” and be­fore they try to make pro­gress, they already de­cide that it is too dif­fi­cult. Did this ever hap­pen to you? And if it didn’t, what is it about your at­ti­tude that pre­vents that?

Prof. Daubech­ies: Well, it is, uh, I tell all my gradu­ate stu­dents that you have to be a bit ornery to work in sci­ence. You have to…I find it im­possible to read a book cov­er to cov­er in math­em­at­ics. Be­cause you start, and you hit a dif­fi­cult proof. And you say, “why do they make it so hard? Come on! There has to be an easi­er way.” And so, you try. And you work, and you work, and you work…and you start un­der­stand­ing why all the dif­fer­ent bits are in it. Well, some­times, you do find an easi­er proof, which is nice. But this gives you an un­der­stand­ing for why…also you have to un­der­stand how math books are writ­ten. Math books are not writ­ten by say­ing, look, we would like to get there, but if you want to do this you would do this and this and this but look, there’s this stum­bling block, so let’s see how we get around it. We al­most find no math books like that. What they do is they fig­ure all that out and then the write it down so that the stum­bling block is sort of swept aside. All these lo­gic­al lines fol­low through and then at the end you get this pol­ished gem and you say, my, how did they find that? Well, they didn’t! They did all the rest, and then they pol­ished it. So, that’s some beef I have with how math books are writ­ten. But that’s an­oth­er thing. But still, you have to be ornery, you have to ask ques­tions, you have to say, “but, come on, wait a minute!” and when you work to­geth­er with some­body it’s a little bit like that too. If someone says to you, “I don’t be­lieve it” you should be able to say, “ah, but this” and so on, and that’s how the whole con­ver­sa­tion goes. But think­ing, “this is too hard for me,” well, we all have that to some ex­tent. I mean, I don’t work on the Riemann hy­po­thes­is.2 I’m just not the kind of per­son who likes to tackle a big thing like that. I do work on things that nobody else has done or that are prob­lems, and there are prob­lems that I haven’t been able to solve and that I set aside and I of­ten work on sev­er­al pro­jects at the same time. So I think it’s ok to think that something is too hard and you’ll have to set it aside and come back at a later time. It’s not ok to think that everything is too hard. I mean, you have to be pre­pared to tackle some things. And if it is too hard, say, “how can I make it sim­pler so that I do un­der­stand it?”

Ken: Do you have a con­fid­ence that if you keep work­ing, you will find new res­ults?

Prof. Daubech­ies: No.

Ken: No?

Prof. Daubech­ies: [laugh­ing] No.

You nev­er know!

Well, we don’t really know where ideas come from. It’s a very strange thing. I think they come from try­ing to re­main wide open to many things. You have to really know your stuff You have to really dig in­to un­der­stand­ing strands of the dif­fer­ent things you’re look­ing at. You should not live in a very nar­row crack between boards. And it’s good to go to sem­inars and it’s good to talk to oth­er people be­cause you have no idea where ideas just fall from.

When my daugh­ter was dia­gnosed with ADD (At­ten­tion De­fi­cit Dis­order), and they told me on what basis they dia­gnosed this, I said, well, I have all those things too. So they asked me some ques­tions and they de­cided I have ADD too. So I have a mild form of ADD…I didn’t know…and when I learned this, I told my daugh­ter, “well, we can stop wor­ry­ing” be­cause what it means is that I have tons of things that flip through my head. First of all, it means you see a whole lot of hu­mor in many situ­ations, which is fun. I think any good car­toon­ist must have ADD of some sort. But, on the oth­er hand, I think it also makes me more cre­at­ive, to see con­nec­tions where there might not be any. And most of these con­nec­tions don’t really ex­ist, but some­times, it shows something…some pat­terns. But you don’t know where they come from, so when you start work­ing on something new, since you don’t con­trol it, how will you know that it’s go­ing to hap­pen again?

After some time you start say­ing, well, it’s happened a couple of times be­fore, so many times be­fore, maybe it will hap­pen again and you say, well, they gave me a job, so they be­lieve it will hap­pen again. But I still, for a long, long time, I felt like a com­plete fake. I felt if people only knew how in­side I was in­sec­ure they would nev­er ever…I mean there was a com­plete gap between the per­son they thought they had in front of them and the per­son who I knew in­side.

Ken: How did you cope with the in­sec­ur­ity then? How did you man­age to keep work­ing and try­ing?

Prof. Daubech­ies: Well, first, you want to keep up ap­pear­ances [laughter]. You don’t want to lose that job! But then, I think that it must be a very rare per­son who does not feel in­sec­ure like that on the in­side. After all, you have a very asym­met­ric way of look­ing at the world. I mean, you are the only per­son that you see from the in­side. Every­body else you see from the out­side. And so from every­body else you don’t see the in­sec­ur­it­ies un­less they have some kind of neur­os­is and really ex­pose them very badly. I think Or­well said in an es­say on Sal­vador Dali…he said that Dali al­ways struck him as com­pletely in­sin­cere in his in­ter­views be­cause he says no per­son feels on the in­side the way Dali pre­tends he feels. On the in­side, you al­ways feel like a whole as­sembly of fail­ures with the oc­ca­sion­al good thing in between. But, first of all, real­iz­ing that every­body feels that way helps. And then second, by want­ing to do re­search…that is to say, find­ing things that nobody else has found be­fore, you’re bound to be on a bad track at times. So when you find something, it’s great! You feel, ab­so­lutely…you feel very, very elated. So I tell my stu­dents, when you find something new you should en­joy [it] for half an hour and then you check the de­tails be­cause it could be a mis­take, but at least you’ve had that half hour of pure joy!

Ken: For half an hour??? Not even an hour?

Prof. Daubech­ies: Well, ok…but you should al­ways check for mis­takes be­cause there could be mis­takes. And if there are no mis­takes, you feel even bet­ter. But it lasts very little. Even if you have no mis­takes, it lasts a couple of days. After that, you have un­der­stood it even bet­ter and you be­gin to feel very stu­pid for hav­ing looked so long be­fore you found it! And you kind of make it part of the tis­sue of math­em­at­ics that you know and at some point it be­comes com­pletely ab­sorbed just like we ab­sorbed decim­al nota­tion in the math­em­at­ics we teach our chil­dren. And at that point, it’s no longer that big joy. It might still be fun to ex­plain it to oth­ers who don’t know it, but that pure thing of “Wow!!!”…that’s over. So you choose a pro­fes­sion where you’re frus­trated a great deal of the time, you don’t know when you’ll find something, and when you find something, you feel, “woah!!!” and then that high is over a couple of days later. So it’s a frus­trat­ing thing…it can be frus­trat­ing…but it’s also a lot of fun. And I like teach­ing also…I like talk­ing about math­em­at­ics to people and teach­ing stu­dents. So it’s not just these wows and dips in between.

Ken: I of­ten feel, I don’t know if you agree, that a large part of do­ing math­em­at­ics is psy­cho­lo­gic­al, but it doesn’t seem that there are any courses in grad school that try to help stu­dents deal with these is­sues.

Prof. Daubech­ies: Yeah, well…I tell our in­com­ing stu­dents al­ways that they have to really work on build­ing a so­cial group…a so­cial net­work around them­selves…that they will learn half their math­em­at­ics from their peers any­way…not from their pro­fess­ors. Just like math­em­at­ics, do­ing math­em­at­ics is a very hu­man pur­suit and it’s very good to do it with oth­ers. I like col­lab­or­at­ing with people. I also like hav­ing my time to fig­ure it out be­cause some­times you have to fig­ure out things in the pri­vacy of your own of­fice be­fore ex­plain­ing it to someone else. So in col­lab­or­a­tions, we typ­ic­ally ex­plain things to each oth­er then we work alone then we get to­geth­er again and so on. I try to foster a lot of get­ting to­geth­er of the stu­dents. I like to give them read­ing courses where I ex­plain for each chapter of the book what it is about and then they have to work through the de­tails and they have to as­sign prob­lems and dis­cuss prob­lems and so on. And then, after they have di­ges­ted that chapter, ex­plain­ing it to each oth­er, worked through the prob­lems, done everything, we go to the next chapter. They find that it teaches them a lot. But it’s also the do­ing it to­geth­er that teaches them. So that is very im­port­ant. So, I try to make sure we have nobody fall­ing through the cracks who is too isol­ated.

If you’re down and you see oth­er people who are not…that helps you. Also, you talk about more things; talk­ing about more things, be­ing ex­posed to more things, leads to more ways out of a prob­lem.

Ken: We have a girl at Girls’ Angle who I per­son­ally think is quite gif­ted. But she has kind of a dis­ease where whenev­er she solves a prob­lem she im­me­di­ately thinks, it must have been trivi­al be­cause she feels that if she can solve it, it must be something very simple. So, she tends to think of her­self as very un­in­tel­li­gent. She’s even said, “I’m so stu­pid”.

Prof. Daubech­ies: Yeah, but, uh, well…I don’t know her, so I don’t know wheth­er it is…it could be the res­ult of something at work that is something in the way girls are brought up…girls are not brought up to feel they are stu­pid, but girls are brought up in a way to try to find com­mon ground among each oth­er, to find ways of shar­ing. And it might be that by say­ing, “I am smart” she feels that she is coun­ter­act­ing…that she is not do­ing the ac­cept­able thing. Or she might even feel that by be­ing smart she is mak­ing her­self too dif­fer­ent from the oth­ers…I don’t know. I really don’t know. I know that played a role for me at some point. At the last ment­or­ing pro­gram we had for wo­men, we had a dis­cus­sion about pro­fes­sion­al in­ter­ac­tions between wo­men and men in the math de­part­ment or, rather, not pro­fes­sion­al, but how col­legi­al in­ter­ac­tions can be dif­fer­ent. And one of the seni­or wo­men said that ju­ni­or wo­men should be aware of the fact, she said, you know when you meet oth­er wo­men…one way in which wo­men bond is that one of them will say something in a funny way but something about a per­son­al weak­ness…something dis­par­aging about her­self. And the way oth­er wo­men will counter will be by do­ing the same thing about her­self. So you’ve shared a weak­ness and that cre­ates a bond so you’re both not per­fect. Well, this is something that you shouldn’t do with most men in the de­part­ment. If you do this with a man, most likely, he could start feel­ing su­per­i­or or he could start ex­plain­ing to you how to solve your prob­lem. [laughter] But this is part of a way of in­ter­act­ing…it’s not a ploy…it’s a tac­tic people use, not con­sciously, but as a way of mak­ing a bridge.

So, I don’t know if it would help, in­stead of feel­ing, “I’m so stu­pid” to think well, “I can do this, but so and so is bet­ter at that,” so as to feel she is not singling her­self out by be­ing able to do the prob­lem. I don’t know. That’s one thing I can think of, but there are oth­ers.

But there’s also the fact that al­ways what you can do your­self doesn’t seem as mi­ra­cu­lous as what oth­er people can do be­cause you’ve seen all the mis­takes that you did on the way to­wards it while the oth­ers just come out with the an­swer. How old is she?

Ken: She’s twelve.

Prof. Daubech­ies: I hope she doesn’t dis­cour­age her­self from do­ing math­em­at­ics.

Ken: Also, she has an is­sue with mak­ing mis­takes. She hates to make mis­takes, to the point where I think it is an obstacle. This is a prob­lem we’re hav­ing at the club: try­ing to con­vince the girls that math­em­aticians make a lot of mis­takes.

Prof. Daubech­ies: Oh yeah!

Ken: Ac­tu­ally, maybe you could ad­dress this?

Prof. Daubech­ies: Oh…everything I start, when I’ve fin­ished, I joke that I’ve done it ten times, be­cause I star­ted and I find that I made a mis­take, so I start over again, and I find an­oth­er mis­take, so I have start over again, and over again! That’s why many of my stu­dents, when they write a pa­per, they write it on the com­puter, I can’t do that. Be­cause even in the writ­ing, I find, it’s not right the way I say that…it’s not ex­actly the way I want it and if I do that on a com­puter, too much of it sur­vives. So I do it in long hand and I erase and so on. I know you can do that on a com­puter, but it is not the same for me.

But, oh yeah, I make mis­takes all the time.

And I have hunches that are wrong. Some hunches are right. But you know, the funny thing is, for one pa­per that I did, many, many, many years ago, I had kept a note­book with every ap­proach that I did…every at­tempt…and many of them led nowhere. And, in the end, I solved that prob­lem and it be­came a pa­per, and I had the im­pres­sion that the in­sight had just come in a kind of flash. I really be­lieve that for many things…and that then, everything had fallen in­to place. And then, I came upon that note­book some time later, and I real­ized that it wasn’t like that at all. I could look in the note­book and see in all those places where it hadn’t worked that there were already some of the germs of the things that did work even­tu­ally. So it wasn’t true at all that I just found it in a flash, in hind­sight. I mean, if I hadn’t made all those mis­takes be­fore, I wouldn’t have had all the build­ing blocks to fit the thing to­geth­er. So all those early things had been ne­ces­sary, I mean, had been part of it. I was really sur­prised, be­cause I had this very dif­fer­ent per­cep­tion of how it happened from what this note­book showed me.

Ken: One thing we’re go­ing to try with the Bul­let­in…there’s a wo­man who just gradu­ated from Smith Col­lege [with a post-bac­ca­laur­eate de­gree in math], her name is Anna Boat­wright. She’s go­ing to solve a prob­lem keep­ing an hon­est re­cord of her at­tempts to solve it. Then she’s go­ing to write an ac­count of her at­tempts to solve the prob­lem, er­rors and all. I think it is very brave of her to do this be­cause I think most math­em­aticians try to sweep all the er­rors un­der the rug be­fore they pub­lish. [Ed­it­or’s note: Anna’s prob­lem-solv­ing was doc­u­mented in a reg­u­lar column in the Bul­let­in called “Anna’s Math Journ­al”. It ran from 2009–2011.]

Prof. Daubech­ies: Well, be­cause that’s the way you have to pub­lish be­cause pub­lic­a­tion is ex­pens­ive. So they try to do it as ef­fi­ciently as pos­sible. And also be­cause they de­vel­op this way of writ­ing with lemma and the­or­em which made it easi­er to check wheth­er something is cor­rect or not. In fact what we do when we write a pa­per is to write it in such a way that some­body who, if all the steps are giv­en, then what they do is give a pro­ced­ure for check­ing that the whole thing is cor­rect. It might be ten lem­mas and you say, “why would they care wheth­er this ob­scure fact is true?” Well, be­cause it is a build­ing block later. And of course, they didn’t find that lemma first, but they give you a way in which you can see that the whole thing is cor­rect. But that’s not the way in which it was found.

Ken: In your opin­ion, should the lit­er­at­ure be a re­pos­it­ory of truth or a place to ex­plain?

Prof. Daubech­ies: I think you need both. That’s one thing…I hope you teach the girls this…stu­dents who come out of high school and who are good at math…I don’t know if they learn this from a teach­er or if it is just the cul­ture, but they be­lieve that you have to fig­ure it out in your head. I mean, you give them a prob­lem, and they stare. My son would do this, and he would say, “I don’t see it”. So I would say, “I can’t help you if all you tell me is ‘I don’t see it’!” They don’t start writ­ing. I mean, when I start a prob­lem, I start im­me­di­ately writ­ing. I start writ­ing what I know, I start writ­ing what I want to get. I write how can I get from one to the oth­er. I start from the be­gin­ning if I know where to start or from the end if I know where I want to go. And I try to con­nect…I try to knit it all to­geth­er. I try to see many ways and hope to build a bridge. Of course, there are people who can do all that in their head, but I think at some point, un­less they are Terry Tao, they are go­ing to hit a place where they can’t do it all in their heads any­more. And if they are Terry Tao, they don’t need me to teach them any­way [laughter]! And when they hit that place, they have to have in place already the tools to handle a situ­ation like that. And the tool to handle a situ­ation like that is to start writ­ing. Put­ting down the dif­fer­ent pieces so that you can look at them and see how it fits to­geth­er. I teach a fresh­man mul­tivari­able cal­cu­lus class which we in­teg­rate with phys­ics. When I do that class I tell them that we are go­ing to go way faster than what they are used to, that they will have to study ma­ter­i­al…usu­ally they don’t study either…they’re used to go­ing so slow that the ab­sorb it by os­mos­is, and we can’t do that any­more. And that they al­ways have to start writ­ing. They have to write what they know, what they want to get, what they know about the kind of prob­lem, and then start see­ing wheth­er they can put the pieces to­geth­er. If you do that, if you don’t be­lieve that writ­ing a prob­lem is go­ing here to…and then writ­ing lin­early to there without a single mis­take and this is the an­swer and I think it is easi­er also if you see it as a whole pro­cess of hav­ing here things, and here things, and here’s some know­ledge, and now we are kind of em­broid­er­ing around it un­til the things start touch­ing, then it might be easi­er to ac­cept that there are go­ing to be in the end lots of things on the page that have noth­ing to do with the solu­tion. But that’s ok, you just se­lect out the part that does.

When I teach, I also have a course that is a first-proof course based in ap­plic­a­tions. I teach styles of proof to the stu­dents. I do it in two-columns. I say, how would we build this proof? I say, we want to get here and so on…and after we have the proof, we then write the proof on the right hand side which is a very dif­fer­ent thing.

Ken: By the way, I’m just curi­ous, you’ve used a knit­ting meta­phor…do you knit?

Prof. Daubech­ies: I used to knit. I don’t any­more. I used to crochet even more. I would love to if I have time, I would love to have a loom for weav­ing. But I make ceram­ics.

Ken: Do you make your ceram­ics with any con­nec­tion to math­em­at­ics?

Prof. Daubech­ies: No…I just like mak­ing things by hand. I’ve been think­ing maybe after I re­tire I’d love to make some art works in which you il­lus­trate some math­em­at­ics.

Ken: So, I’m go­ing to ask you a com­pletely dif­fer­ent ques­tion. I’ve been want­ing in­ter­viewees to ad­dress this but many are un­der­stand­ably re­luct­ant to. Do you think there is gender bi­as in the field of math­em­at­ics today?

Prof. Daubech­ies: [pause] There cer­tainly are still in­stances that I hear about. If you ask any young wo­man, she has stor­ies. Among the gradu­ate stu­dents I think they’ve en­countered it more among their peers than the pro­fess­ors. Some of the pro­fess­ors are not ne­ces­sar­ily very friendly to­wards gradu­ate stu­dents…it’s not that they are not friendly, it’s just that they are ob­li­vi­ous in a kind of gender neut­ral way…but I think it might be re­sen­ted more by wo­men. But def­in­itely there are people who think yes, who kind of as­sume that a wo­man will not be as good and who may even ar­tic­u­late that. Usu­ally, they are more per­ceived as jerks, but it ex­ists. And then, def­in­itely, if you look at stat­ist­ics, or, well…the stat­ist­ics are com­plic­ated by many, many is­sues. Be­cause wo­men also un­for­tu­nately drop out more from the field at trans­ition points than men and so we want to do something partly be­cause the num­bers are low, I think. So, I think…I haven’t seen many cases of blatant dis­crim­in­a­tion, but I do think there still is some. I think it’s at the level where it’s just a con­sequence of the fact that there are much few­er wo­men than men in the pro­fes­sion. I mean, people make im­pli­cit as­sump­tions based on what they see.

Ken: There is an ex­plan­a­tion put forth by some psy­cho­lo­gists that the low rep­res­ent­a­tion of wo­men in the up­per levels of math­em­at­ics is not due to gender bi­as but is due in­stead to dif­fer­ences in pref­er­ences between what girls like and what boys like. What do you think of that ar­gu­ment? Do you think that it is val­id?

Prof. Daubech­ies: I think that it does not ex­plain why more wo­men than men leave the field. I mean, these wo­men already liked math­em­at­ics to be­gin with and then con­cluded it is harder for them to stay in and com­bine a fam­ily and ca­reer, to make their mark…whatever…so I mean, I think I have no a pri­ori reas­on to as­sume that…I don’t be­lieve in it be­cause I’ve seen plenty of wo­men very in­ter­ested in math­em­at­ics. And I’ve seen wo­men tal­en­ted in math­em­at­ics and who did leave the field usu­ally be­cause they don’t like a job where the ten­ure clock is ra­cing at the same time as rais­ing a fam­ily and things like that. An­oth­er thing is that many math­em­aticians seem to be­lieve that, well, there are two things. First of all, many math­em­aticians who are con­sidered to have left the field, for in­stance by the AMS, are still do­ing math­em­at­ics in a broad­er sense…in the sense that they are ap­ply­ing very much their ana­lyt­ic­al skills and their pre­cise reas­on­ing skills in their new jobs and they find that math­em­at­ics trained them very well for that, and they en­joy it. So, we haven’t really “lost them”. The second thing is that so many people in aca­demia seem to feel that it’s al­most a fail­ure not to go to aca­demia. For your girls, that’s not an is­sue, they are still young, but I think that is ab­so­lutely so ri­dicu­lous. Even people who agree that it is ri­dicu­lous still have this im­pli­cit as­sump­tion. If they have a very good stu­dent who then de­cides to do something else, they say, “oh, such a pity, he could have been!” But what does it mean? It still means you have an im­pli­cit value sys­tem…I mean, [the thought that] “only the people that couldn’t have been do something else”…which is ri­dicu­lous. Math­em­at­ics is a skill that is very fun­da­ment­al that many people have and we need many people who have it in spades, and we need it every­where, and we should be happy to spread it around.

But, we do lose wo­men more than men. I think it is partly be­cause, well, the track it­self is kind of hard for wo­men if they want to raise a fam­ily.

Ken: One of the goals of Girls’ Angle is to be­come a com­munity of sup­port for all wo­men en­gaged in the study, use or cre­ation of math­em­at­ics. One of the is­sues I think about is this is­sue of rais­ing a fam­ily. I’ve of­ten wondered, you know, every per­son has two par­ents…why does it seem that the bur­den is on the moth­er to do the child rear­ing…why is it that when a male math­em­atician has a child there is no stigma as­so­ci­ated with wheth­er he is still go­ing to be able to do the math?

Prof. Daubech­ies: Well, yeah, well, I don’t know. Well, part of it is that when the baby is really little, the moth­er is nurs­ing and that is something only the moth­er can do. But, yeah, I agree. So ac­tu­ally, when gradu­ate stu­dents and postdocs ask me how do you com­bine a fam­ily and a ca­reer, I tell them that the first thing to do is to choose their hus­band well. It may seem a bit silly. But it is very im­port­ant that you dis­cuss ahead of time how you see things go­ing in a fam­ily and in some de­tail so you don’t have vague ex­pect­a­tions. Well, I’m hap­pily mar­ried now but earli­er I had a re­la­tion with a man who was a very nice man but I think he ex­pec­ted something dif­fer­ent from his fiancé than what he ex­pec­ted from his wife, which can be very frus­trat­ing for a young wo­man to meet a man who really ap­pre­ci­ates you be­cause you are in­tel­lec­tu­al, be­cause you have all these in­ter­est­ing pur­suits in life and so on, but then ex­pects that after mar­riage that you will take care of the house and cook­ing and…which wouldn’t leave you any time any­more to be the per­son that they liked be­fore. I had im­pli­citly as­sumed that since he liked these things, we would work to­geth­er, that we both would have the time left to be the per­sons we had been. And it didn’t work that way. But, we hadn’t dis­cussed it. And it took me a while to fig­ure out that was what was wrong. So I tell them, that’s im­port­ant.

The sup­port of a hus­band who views it in the same way is ab­so­lutely im­port­ant. But after that, there’s still the rest of the world and people may find it strange that this couple works in a dif­fer­ent way from most couples they’ve seen. So, I think hav­ing a sup­port group like that would be great.

Ken: Do you have any sug­ges­tions for how this can be im­ple­men­ted?

Prof. Daubech­ies: Ummm, I don’t know…I mean I know that I’ve had sup­port groups with wo­men and we’d hang out, and we’d get to­geth­er and chat…so, def­in­itely have a corner, I don’t know how big your fa­cil­ity is…but hav­ing a corner where people can just chat, or even a sep­ar­ate room, where they can chat and not be over­heard by every­body else work­ing on math would be good.

Ken: Someday we hope to ex­pand the age range to in­clude tots and have a day­care cen­ter over­seen by people who love math. So moth­ers could…

Prof. Daubech­ies: Oh, that would be very good.