Okay, so obviously I would like to start by thanking Ursula not only for organizing
this amazing meeting, but also for drawing me into this project in the first place.
As you can see this, have I put it there, yes.
This is joint work with Ursula and also with Adrian Rice,
who also deserves special thanks because he was delegated to
write the first draft of our paper on this subject.
And this landed in our inboxes just a few weeks ago, a near perfect
first draft which has been fantastically useful when writing this talk.
Okay, so my title is The Mathematical Correspondence of Ada Lovelace And
Augustus De Morgan, I'm using that as
a way of trying to gain a proper assessment of
how Ada was as a mathematician, just by focusing on this correspondence.
And I'll explain how the correspondence came about, a bit later on.
So, there have been various, different assessments of Ada, as we all know.
We've heard about different ones over the last 24 hours.
She's perhaps hailed as being a visionary in computer science and
it seems to be that, that is then carried over to the mathematics.
It's claimed that she was a brilliant mathematician also.
And I'm not quite sure that's the case.
And I will sort of make my argument as I go along.
So of course, broadly speaking, the previous assessments of
Ada's abilities both generally, both in computer science and
in mathematics, fall very broadly into two camps.
[LAUGH] So, we have some very occasionally extravagant claims on the one side,
and we have the backlash against that on the other side.
And as you go through you can perhaps see why these things have come about, so
maybe if I have time I'll comment on why I think these views have emerged.
But our purpose in our work has been to actually, to provide for
the first time, because what I should say is the people who expressed these views
haven't always gone into the mathematics in fantastic detail.
So, this has been our goal to actually get into the archives and
have a look and just look at what math is there.
So, the idea was to provide a sober,
objective assessment of how Ada was as a mathematician.
So, we just heard a little bit about her mathematical education.
Sorry, I'm skipping ahead.
The truth is gonna be somewhere in the middle.
This is gonna be the point of the talk.
I'm going to sort of, I'm going to argue why she wasn't brilliant,
but why she wasn't stupid, and I'm gonna end up in the middle somewhere.
So, we've heard a bit about her early education.
Surely, she was learning arithmetic with tutors quite early on.
Then, a bit later she started to learn Euclid, as any mathematical education
of any depth in the 19th century would begin with Euclid.
And she was tutored by various people.
Dr. King, a friend of Lady Byron.
There was William Friend, a mathematician, who had himself tutored Lady Byron.
There was Mary Summerville, occasionally.
There were letters exchanged there.
So, all of these people are tutoring Ada mostly at a distance.
This is mostly through letters.
Which is why we know about it, because the letters that are there in the body.
And early on, Ada's interest in mathematics seem to have been
as a means to an end.
She wanted to understand more astronomy.
She wanted to understand more optics.
And she says this explicitly in a letter to Dr. King.
But as it goes on, she does seem to get more absorbed by the mathematics for
its own sake, these hints of applications drop away, to some extent anyway.
And so, by the mid-1830's, she seems to be quite confident
with the mathematics she'd learned with the Euclid in particular.
So much, so that she set herself up as a tutor to a young family friend,
Annabella Acheson, and there are some enormous letters
proving propositions from Euclid for Annabella's benefit.
What's not clear to me is whether Annabella actually wanted to be tutored in
Euclid.
[LAUGH] So then, that's as far as things went before Ada married.
So, 1835, she married.
Three children followed.
And then, we get to 1839, and Ada is again
wanting to study more mathematics to go beyond what she's already done.
So, she's looking around for a tutor.
She certainly asked Babbage to help her find someone.
But it doesn't seem to have been through Babbage, that she came to
Augustus De Morgan, the founding professor of mathematics at UCL.
He and Babbage had been friends for many years, but the connection to Ada seems to
have come through De Morgan's wife Sophia, who was a friend of Lady Byron.
And so, Ada and De Morgan entered into what was essentially a correspondence
course in mathematics.
And we have quite a lot of that surviving in
Box 170 of the Lovelace-Byron papers down at the end.
There are mathematical sheets scattered throughout the various boxes.
But Box 170 has a particular concentration of them.
So, 357 sheets in total.
The first 43 comprised 20 letters from De Morgan to Ada.
The next 120 or so are the 42 letters in the other direction.
And in the second half of the box is just assorted mathematical jottings.
And a very large part of this doesn't seem to be relevant to this project.
We think because of the handwriting, because of various references to
textbooks that haven't been published by this stage, we think a lot of these papers
are actually Ada's daughter's from when she was learning mathematics.
But there are other things in there, in these assorted mathematical jottings,
there are enclosures that were sent with the letters, and also,
the Konigsberg British Sheets we've seen already, it is in there.
Okay, so what was the nature of this correspondence?
Course well it was a largely self-motivated one, De Morgan was
there to help, but Ada was mostly working through De Morgan's textbooks.
He was directing her to appropriate reading and
I'll show you some examples of that in a moment.
There are some gaps, but I think anybody who's ever tried to study something on
a self-motivated basis knows that there will always be gaps because you get
sidetracked by things and have to come back and restart and so on.
And of course there are lots of letters going back and forth.
Ada's asking about such-and-such a thing that she doesn't understand and
De Morgan writes back and explains, and Ada writes back with a new question,
and sometimes more than one letter per day.
And occasionally, they were meeting face to face, and
we see this littered throughout the letters, we'll see things like this
from De Morgan to Ada, we'll be happy to see you on Monday evening and
Lord Lovelace too if he not be afraid of the algebra.
[LAUGH] So, this conjures the nice image I think of Ada and
De Morgan sitting at a table doing mathematics.
Lord Lovelay's I don't know,
perhaps he has to amuse himself in the corner, I don't know.
But very similar to the Kenigsberg British Sheet where we
have the pencil going through the ink and we have a picture of Babbage and
Ada sitting at a table again, doing mathematics.
So, this is the sort of reading that Ada was doing,
these are essentially sort of elementary textbooks by De Morgan.
Or elementary as compared to the other things that Ada was doing.
Because up to this point,
she hadn't had a particularly systematic mathematical education.
There were some gaps in there.
And so, you see in the letters that De Morgan keeps having to send her back to
these books.
She wants to learn calculus, that's the main thrust of what she was doing.
But it keeps turning out she doesn't have the necessary algebra,
she doesn't have the necessary trigonometry,
so De Morgan keeps sending her back to these books to fill these gaps.
Other books she used, well, we have Peacock's Treatise of Algebra.
Peacock was De Morgan's Cambridge tutor.
So, this is a book from 1830.
So, by the time this correspondence was going on in 1840-41,
this book had become quite difficult to get a hold of.
But we know that Ada managed to get it because she wrote to De Morgan
with an element of shock that she had to pay 2 pounds, 12 shillings,
and 6 pence for this book, which originally sold for 30 shillings.
So, she was quite shocked by this.
On the right hand side, we have a page from the Penny Cyclopedia,
which is a publication that was produced by the Society for
the Diffusion of Useful Knowledge, with which De Morgan was involved.
And he wrote lots of the mathematical articles for this encyclopedia and
this is a page from the article on negative and impossible quantities.
Impossible quantities meaning complex numbers in modern terminology.
And this is something I'll come back to in a bit.
But the main text that Ada was using was De Morgan's calculus, because as I said,
calculus is what she really wanted to get into.
Okay, so that's an overview of what she was doing.
Now, there is the question of how she was doing,
how she was doing as a mathematical learner?
So, what I've done is I've put together a few indications of her weaknesses and
then, I'll follow that with an indication of her strengths.
And hopefully we'll come to some kind of conclusion, however vague, at the end.
Okay, oh yes, so this is what I was just commenting on.
So this is the first of her weaknesses,
the weakness that was rectified fairly quickly, was
just the fact that she had insufficient foundations to study the calculus.
So we find comments like this.
You must make up the points in algebra and
trigonometry that you have left behind and Ada did.
She was very impatient to get on with things,
but you do eventually see that she acknowledges the need to do this.
She said, my algebra wits not having been stretched proportion,
not having been quite stretched with some of my other wits.
But you see things in this correspondence where
there are things that perhaps she ought to know by this stage but she doesn't,
because she's never been taught it.
So, here's an example of something that she had some difficulty with.
This is a concern with the equation of a curve and we can see it there,
it's y equals x squared.
And she really struggled to understand what do we mean
by the equation of a curve?
And this is I suppose of GCSE level type thing now, but she said,
well what is the equation?
Is it the sequence of values that we get from this, so
up the scale there, or is it the spikes that we have or what?
And she doesn't get it, and there's a sort of back and
forth where De Morgan is really trying to explain, well, we just mean this.
This is the curve and this is the equation and you put values in.
And she did seem to just, I don't know, it's odd
to read because she just seemed to label the points lightly, I don't know why.
But this is a particularly nice example of some of the papers.
Because at the top here we have Ada's,
she's written out her understanding of what this means.
For De Morgan to correct, and then, here in a slightly darker ink and
different handwriting is De Morgan's corrections.
His comments on her understanding,
trying to make her see where she's misunderstanding.
And there are similar such things certainly in the earlier letters.
There's a very similar exchange about logarhythms she doesn't understand
logarithms and that again, takes a few exchanges for her to pin down the idea.
But these problems generally seem to have been sorted out in the long run.
They perhaps in some cases take a bit longer to resolve than you would perhaps
expect, but they do tend to be resolved.
One thing that isn't really ever resolved, as far as I can tell, is the struggles
that Ada had with algebra, just simply manipulating symbols on the page.
She just frankly wasn't very good at it.
So, here we have November 1835, she's writing to Mary Somerville for
asking her for help, manipulating some trigonometric identities, and
you can see it there, I think.
Cosine A equals sine A minus B, etc.
And we have a nice little exchange of letters there.
But this is 1835,
so this is still comparatively early in Ada's mathematical learning.
She hadn't done a great deal up to this point, so
it's perhaps understandable that she's having this difficulty at this point.
But it persists.
This is five years later.
Ada confesses to De Morgan in a letter that she's completely
baffled by this problem showing that the thing at the bottom
satisfies that equation at the top for all values of A.
And this is something that never really goes away.
This struggle with algebra, and
in fact this is a sheet from which we had a quotation from yesterday.
These functional equations meaning the things that make the equation at the top
5x plus y, etc., are complete will-o'-the-wisps to me.
And like I said, this just continues throughout the correspondence.
This sheet, incidentally, is one of the ones on display in the Bodleian,
so if you haven't already seen it, you can go and have a look and
see if you can understand what it is that Ada is struggling with, more specifically.
This type of comment also,
I should say is an example of her very whimsical turn of phrase.
So, she's whimsical, she can also be quite long-winded and I wonder if that's perhaps
one of the reasons why people have dismissed her in the past.
They haven't been perhaps necessarily been prepared to work past this,
her way of expressing herself which can be a bit exhausting at times to read it,
but I don't know if that's the case.
Perhaps, it's also worth mentioning here, and I didn't put this on a slide because
it didn't look particularly interesting on a slide,
but there are issues with dating of these letters.
Ada was appalling at dating things,
she was very slap dash with the dates.
There are, for example, two consecutive lessons, I can't remember what the date on
them is exactly, but it's something like Sunday, the 6th of June, 1841.
Followed immediately by a Monday the 6th of June 1841.
And then, we look at a calendar, you find that the 6th of June 1841 was a Tuesday.
[LAUGH] So, you're never quite sure whether to believe the dates completely,
if she put them on at all.
But her son Ralph sorted through some of this material after her death and
put some dates on them.
But we don't think he's got it quite right.
I'm not sure if it's this letter but
one in this sequence has been assigned to 1842.
So, it looks like we have Ada doing this
material on function of equations in November in 1840 and
then still by November of 1842 she's asking elementary questions about it.
Well, the reality is we think that this is not 1842, this is 1840, so suddenly
it seems a lot more reasonable that she's asking these questions about it.
So, that's the sort of detail that interests me, but
I don't know whether it would work so well in a slide.
And I've already mentioned another of her weaknesses, impatience and
occasionally over ambition so she wishes she could go on quicker.
She's very disappointed to sit down and find an hour or
two later she has accomplished one-twentieth part of one's intentions.
I think we've all had days like that.
So, you find De Morgan reigning her in, as well as telling her, no,
you've got to go back and fill in these gaps in the algebra and trigonometry.
You should never estimate progress by number of pages.
And it's interesting to see this comment, because she was saying precisely the same
thing to Annabelle Agerson in her tutorship of her just five years earlier.
So, she's not heeding her own advice here, but
she does sort of settle into things eventually, it does seem.
Right so, so far I've been quite particular.
I've shown you some examples of where I think Ada's weaknesses were,
so now i want to, on a more positive note turn to her strengths.
And this is slightly problematic because I think, as Betty Toole commented yesterday,
the letters to De Morgan are letters about what Ada doesn't understand.
If she understood something then, she wouldn't write about it.
So, there's a certain amount of extrapolation, and
dare I say speculation involved in saying what she was good at,
which I think is why some authors have, in my view,
gone slightly too far on occasion.
But anyway, so the strengths that I think she had were, first of all, the thing I've
already mentioned, the self-motivation, so she'd corresponded with Dr. King,
with William Friend, with Mary Somerville, so much of her higher mathematical
learning had be done at a distance and this pattern continues with De Morgan.
So, she must of been determined to do this just to keep things going.
And speaking for myself, I've not always managed to keep these sort of things going
when I've tried to study things myself.
She's determined to understand every last detail of things,
it's something that you notice.
As you work through the letters you get to the point where you think oh no not
that integral again.
But she wants to understand every detail.
And she doesn't want to just apply rules, she wants to understand where the rules
are coming from and so, as I've said here I need to understand why rules work,
so, for example, she's presented with the naive manipulation of differentials and
she wants to know well why does that work?
Why do we write dx on the end of everything and
if we have dy by dx then, why could we simply multiply by dx?
Why doesn't it work like that?
So, she wants to understand.
And there's also some scepticism over existing rules.
There was a principle which held amongst British mathematicians at this time.
The principle of the permanence of equivalent forms which said that anything
you can do in arithmetic, you can apply to other contexts.
We don't hold to this rule anymore because well, we know it's not true.
But this was presented to Ada just as a rule that she could apply and
she wanted to know why.
Well, why does this work?
And she doesn't seem to have ever got a satisfactory explanation.
At least not in the papers that survive.
Perhaps because she can't give a satisfactory explanation.
She also has a very critical eye.
There's a lot of letters where she's pointing out typos in the textbooks.
She's not right all the time.
Sometimes it's her mistake and
she hasn't seen it but she has quite a good hit rate with these things.
She's really understanding what's going on and
getting into the details and seeing where problems are.
And this is where I get a bit more speculative, I'm afraid.
I think, she had a broad view of mathematics.
We've seen that a bit with the comments on her computer science,
that there's a question about what she actually contributed to this paper,
but she seems to have had this big vision of what computing might do.
And so perhaps, she wasn't good at algebra.
Perhaps, she wasn't good at the details but
she does seem to have had a good broad view of mathematics.
A view towards generalization possibly, I've got an example coming up.
It's a feature of the mathematical minds to prove a theorem and
then seek to generalize it, so she has an awareness of that process.
And possibly also, there's some research awareness, and there's a letter to Mrs.
De Morgan, where she comments.
Oh, I understand that Mr. De Morgan is working on such and such.
De Morgan, was sending at least on one occasion,
was sending her a copy of one of his research papers.
Now, we've no idea whether she read it, whether she understood it,
what she thought of it.
But the fact that he was sending it, makes you think, well, you know?
Maybe he thought that she will get something from this.
So, my example for generalizations, then, is this here.
This is a much quoted passage in connection with Ada,
and this comes after she's just been reading the article I showed
you from the Penny Cyclopaedia, the negative and impossible quantities.
So, she's read that, but she's made a few comments in this letter.
And it's about complex numbers which, amongst other things,
can be used to do geometry in two dimensions.
And she's musing that maybe we can do something
to enable us to do Geometry in three dimensions.
And this is cited as her vision, because this is just two years before
the Irish Mathematician William Rowan Hamilton did just that.
However, if you look at
the Penny Encyclopedia article you see that actually everything up to
the last comma in this quotation is more or less there in that article.
So, is she just parroting what she's read?
It just seemed to be that case.
So then, it all hinges on what we make of this last bit.
This and so on, ad infinitum possibly.
Is that naive speculation?
Is it genuine insight?
I don't know and I don't think we have any way of knowing.
So, I'm just gonna leave that dangling.
This is just sort of a matter for debate, I suppose.
So, just to bring things to a close, then.
I'm going to quote from another much quoted source in connection with Ada.
This is a letter that De Morgan wrote to Lady Byron in 1844
saying what he thought of Ada as a mathematician.
So, he's commenting here that her abilities are so utterly out of the common
way for any beginner which is a slightly odd compliment.
Is he saying that she's good, or is he saying that she's good for
a beginner, I don't know.
At least in that paragraph anyway, but he does go on and
it does seem to be quite complimentary.
He comments the tract about Babbage's machine is a pretty thing enough, but
I could I think produce a series of extracts out of Lady Lovelace's first
queries upon a new subject which would make a mathematician see that
it was no criterion of what might be expected of her.
So, he thinks that she has more in her, that she doesn't really seem to have had
a chance to pursue.
And so, I've tried basically to do this,
to produce a series of extracts from Ada's correspondence with De Morgan.
And what I hope I've demonstrated is it's a bit of a stretch to say she was to say
she was a mathematical genius.
She was certainly competent, definitely competent in most things.
And so, she certainly wasn't stupid.
This is definitely not true.
So, to go back to my diagram at the beginning,
she certainly does lie somewhere in the middle, and
I think that's what makes the question interesting.
Thank you. [APPLAUSE]