Ada, the Enchantress of Numbers:Poetical Science
Page 13
To Lady Byron
Tuesday 5’o clock, 15 August [1843]
St James’ Square
. . .No one can estimate the trouble & interminable labour of having to revise the printing of mathematical formulae. This is a pleasant prospect for the future, as I suppose many hundreds & thousands of such formula will come forth from my pen, in one way or another.
You will receive a few copies (amongst a hundred that are printed separately for me). But where and how should I send them? . . .
I am uncertain as yet how the Babbage business will end. He has written unkindly to me. For many reasons however, I still desire to work upon his subjects & affairs if I can do so with any reasonable prospect of peace. I have written to him therefore, very explicitly; stating my own conditions, without which I positively refuse to take any further part in conjunction with him, upon any subject whatever. He has so strong an idea of the advantage of having my pen as his servant, that he will probably yield; though I demand very strong concessions.
If he does consent to what I propose, I shall probably be enabled to keep him out of much hot water; & to bring his engine to consummation, (which all I have seen of him & his habits the last 3 months, makes me scarcely anticipate it ever will be, unless someone really exercises a strong coercive influence over him). He is beyond measure careless & desultory at times. – I shall be willing to be his Whipper-in during the next 3 years, if I see fair prospect of success. Much of this is W’s suggestion; (altho’ W – thinks B’s conduct to me has recently been very blamable).
William was very supportive of Ada’s publication. She wrote to her mother that William considered that her work placed him “in a far more agreeable position in this county.” She continued by exclaiming: “Oh dear! how mercilessly he carried off my proofs & revises to some of his friends who came here; despite my remonstrances as to their blotted & unintelligible state!” She had good news. “Babbage & I are I think more friends than ever. I have never seen him so agreeable, so reasonable, or in such good spirits!”
Babbage replied to Ada’s request, in probably the most often quoted letter:
9 September 1843
My Dear Lady Lovelace. I find it quite in vain to wait until I have leisure so I have resolved that I will leave all other things undone and set out for Ashley taking with me papers enough to enable me to forget this world and all its’ troubles and if possible its’ multitudinous Charlatans—every thing in short but the Enchantress of Numbers.
My only impediment would be my Mother’s health which is not at this moment quite so good as I could wish.
Are you at Ashley? and is it still convenient with all your other arrangements that I should join you there? – and will next Wednesday or next Thursday or any other day suit you: and shall I leave the [?] road at Taunton or at Bridgewater and have you got Arbogast Du Calcul Des Derivations with you there at Ashley. I shall bring some books about that horrible problem – the three bodies which is almost as obscure as the existence of the celebrated book “De Tribes Impostoribus.” So if you have Arbogast I will bring something else.
Farewell my dear and much admired Interpretess.
Evermost Truly Yours
C Babbage
Philosopher’s Walk at Ashley Combe
Poetical Science
The collaboration between Babbage and Ada ran into serious difficulties.
1. What were the differing perceptions of their goals?
2. How did Ada deal with that?
3. What did vanity have to do with their difficulties?
4. How did they finally come together again?
15
The Analyst and the Metaphysician, and the Analytical Engine:
A Selection from Ada’s Notes
[1843]
Mechanics is the paradise of the mechanical sciences
because by means of it one comes to the fruits of mathematics.
Leonardo DaVinci (1452-1519)
I am a philosopher. Confound them all –
Birds, beasts and men: but no, not womankind. –
Lord Byron quoted by Babbage on the title page of his autobiography, Passages
Since I wrote this chapter back in 1991, so much has changed in the computer world. The origin of the concept of the computer has become more remote. However, when you understand the concept, the source, it becomes easier to predict what happens in the future. At the end of this chapter, the poetical science activity deals with how you can help determine the future.
Babbage’s idea for the Analytical Engine was the conceptual birth of the computer revolution. There were four components to the Analytical Engine—Input, storage, processing, and output. The input medium was the punched card. There were two sets of cards—variable and operational cards. Though there was no programming in the modern sense, by arranging the cards one could program the engine to perform a repeating cycle or process, taking numbers from the store. Babbage planned to store over 1000 fifty-digit numbers. The numbers were then processed in what Babbage called the “mill.” In Babbage’s Passages he stated that there were three mechanisms for the output of numerical information: an apparatus for printing on paper, a means for producing a stereotype mould of the tables or results it computes, and a mechanism for punching on blank pasteboard cards or metal plates the results of any of its computations.
When I started doing research about Ada in 1984 I heard rumors that Ada did not write the first table of instructions since Babbage had already written previous programs. These rumors spread and were mentioned in science fiction novels, popular magazines, and many academic books. The allegations were never footnoted. I finally found Babbage’s former tables and examined them. They were very simple tables, not as complex as the Bernoulli numbers that Ada suggested. Babbage many years later in his book Passages wrote that he completed the table of Bernoulli numbers, but Ada wrote that Lord Lovelace was inking them over. If Babbage was checking the table he missed a bug, a very easy one to catch. Can you find the bug?
In May 1999 I co-authored with Eugene Kim an article for Scientific American entitled “Ada and the First Computer.” It was translated into many languages. In France it appeared in July 1999 in Pour La Science. The German translation in Spektrum Der Wissenschaft would have delighted Ada as she had always wanted to write for German publications. Ada does deserve to be considered at the forefront of the computer revolution; the evidence supports that view.
Since I have a Google alert for the title of my book I get many hits everyday. Ada Lovelace Day is celebrated every 24 March when many people throughout the world write blogs about Ada. On 28 May 2010 Jonathan Strickland did a wonderful blog: http://blogs.howstuffworks.com/2010/05/28/ada-lovelace-and-techstuffs-200th-episode/
A video on You Tube underscores the theme of my first article in the Byron Journal in 1987 and the theme of this book, Poetical Science. I think it is great, and hope you enjoy it as well: http://www.youtube.com/watch?v=g0Kq85fQLWE.
Babbage covered thousands of pages with designs and diagrams. In that form, it would be difficult to absorb the information or support the building of a prototype. To describe how a system that had not been built would work is very difficult (that essentially was done in Menabrea’s description). To relate that description to its value and its use and abuse is a difficult and critical task as well, e.g., having had in 1940 a clear statement describing the power of atomic energy, its potential use and abuse, and having that statement be meaningful today. Ada’s Notes are still thought–provoking over 150 years later, especially in light of the interaction between the computer and nuclear power and our desperate need to know not only technology, but the ramifications of technology.
Babbage supplied the concept and design, and Ada, being both an analyst and metaphysician, put that concept and design in an appropriate context on both micro and mega levels. She asked critical questions that many of us who are not professional mathematicians, scientists, or software engineers would ask. By asking those questions
she has given us a methodology and a language to understand the content and concept of a technological innovation. She integrated what we now refer to as digital or scientific skills of reason and analysis, from verbal to numerical, with the poetical skills of imagination and metaphor.
Ada’s Notes foreshadowed the capability of the modern computer and the impact such a development would have on the language of science. The Notes were written for an educated Victorian audience and are probably not of interest to many people today. Those who are interested can read the original, which now has been reprinted in several books. The selection presented here highlights some of the issues discussed at the time and how they relate to the modern computer and to the software language Ada. (To differentiate Ada Lovelace from the software language Ada, the latter is italicized in this chapter). It is important to mention once again that this chapter, the annotation of the excerpts from Ada’s Notes, is collaboration between Colonel Rick Gross, United States Air Force, and me, and represents our views.
All quotations and page numbers refer to the original Memoir, which was printed in Scientific Memoirs, Selections from The Transactions of Foreign Academies and Learned Societies and from Foreign Journals, edited by Richard Taylor, F.S.A., Vol III London: 1843, Article XXIX. Sketch of the Analytical Engine invented by Charles Babbage Esq. By L. F. Menabrea, of Turin, Officer of the Military Engineers [From the Bibliothèque Universelle de Génève, No. 82 October 1842]. The whole article can be found at http://www.fourmilab.ch/babbage/sketch.html
Ada added a footnote to her translation. She emphasized the difference between Pascal’s machine, which can be compared to a calculator, and Babbage’s Analytical Engine, which can be compared to a modern day computer. Ada translated what Menabrea wrote: “For instance, the much-admired machine of Pascal is now simply an object of curiosity, which, whilst it displays the powerful intellect of its inventor, is yet of little utility in itself. Its power extended no further than the execution of the first four operations . . .” Ada augments Menabrea’s statement and clearly defines the boundaries of Babbage’s Analytical Engine.
From Ada’s footnote to her translation of Menabrea’s work on p. 670:
This remark seems to require further comment, since it is in some degree calculated to strike the mind as being at variance with the subsequent passage (page 675), where it is explained that an engine which can effect these four operations [+,-.×, ÷] can in fact effect every species of calculation. . . The explanation lies in this: that in the one case the execution of these four operations is the fundamental starting-point, and the object proposed for attainment by the machine is the subsequent combination of these in every possible variety. . . The one begins where the other ends. . .
On p. 687 Ada made a mistake in the translation that was not caught by either Babbage or Wheatstone, who, as can be seen from Ada’s correspondence, were supposed to be proofing the translation and Notes. Ada translated “cosine” incorrectly. However, Herschel Babbage, years later, corrected the mistake with another mistake, 1/0. While others have focused and made an issue of a minor typo, we would rather focus on what Ada did do.
In her first “Philosophical Note A,” which Babbage liked so much, Ada defined the boundaries of the Analytical Engine, and the details of how the Analytical Engine would perform its tasks. She emphasized that the Difference Engine, Babbage’s first calculating engine, was designed primarily for calculating and printing tables, but the Analytical Engine was a mechanical and conceptual leap.
From Ada’s Note A, p. 691
The Analytical Engine, on the contrary, is not merely adapted for tabulating the results of one particular function and no other, but for developing and tabulating any function whatever. In fact the engine may be described as being the material expression of any indefinite function of any degree of generality and complexity. . .
Ada had now emphasized the fundamentally different capability of the Analytical Engine, that is, to be able to store a program (a sequence of operations or instructions) as well as data (informational values themselves). At this point, she began to recognize and to amplify the increased responsibility this new capability placed upon the machine’s user, to specify the stored program both precisely and in complete accordance with the user’s interest. Her recognition of this increased responsibility is a remarkable insight, in that the magnitude of this specification task (a task we refer to today as software development) is only now being appreciated.
It is accordingly most fitting that the programming language Ada, developed in the early 1980s by the U.S. Department of Defense, provides the most precise facilities for this software development (specification) task of any general–purpose software language for large–scale problems existing today.
In the following passage, Ada explained the difficulty of the software development task, that is, the difficulty of communicating to the machine what it is we expect it to do. But note that in so doing, she also extolled the power of mathematical language when it is precise. Thus, a software language capable of great precision in specification (like the Ada language) also provides great power.
Indeed, throughout this translation one is struck by the appreciation Ada exhibited for the principle that power comes from disciplined creativity. Neither her analyst nor metaphysician persona is allowed to overcome the other, resulting in a synergy of amazing potential. The powerful, innovative Ada software language is a fitting namesake. Its advanced features have been implemented in a framework that encourages their use in a structured, repeatable software engineering process. Indeed, Dr Frederick Brooks, a renowned modern software authority, forecasted that the Ada language’s “greatest contribution will be that switching to it occasioned training programmers in modern software-design techniques.”
From Note A, p. 693
The confusion, the difficulties, the contradictions which, in consequence of want of accurate distinctions in this particular, have up to even a recent period encumbered mathematics . . . It may be desirable to explain, that by the word operation, we mean any process which alters the mutual relation of two or more things, be this relation of what kind it may. This is the most general definition, and would include all subjects in the universe . . .
They will also be aware that one main reason why the separate nature of the science of operations has been little felt, and in general little dwelt on, is the shifting meaning of many of the symbols used in mathematical notation. First, the symbols of operation are frequently also the symbols of the results of operations . . .
Secondly, figures, the symbols of numerical magnitude, are frequently also the symbols of operations, as when they are the indices of powers [e.g., 2 and 32]. . . [In] the Analytical Engine . . . whenever numbers meaning operations and not quantities (such as indices of powers), are inscribed on any column or set of columns, those columns immediately act in a wholly separate and independent manner . . .
One of Ada’s first letters to De Morgan, which does not remain, dealt with acoustics. De Morgan directed Ada to the Penny Encyclopedia to find out more about the relationship between mathematics and music. Ada’s great love was music and she speculated how the Analytical Engine might deal with it. We have highlighted in bold type sections which have been quoted often. For example, guest editor Denis Baggi’s introduction to the July 1991 issue of Computer (devoted entirely to computer-generated music) began with an acknowledgment that Ada was the first to have suggested such an application.
From Note A, p. 694
Again, it [the Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine . . . Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate an
d scientific pieces of music of any degree of complexity or extent.
Once Ada had made the distinction between numbers and the operations to be performed, it was not difficult for her to project further how the Analytical Engine would then be capable of giving two types of results—numerical and symbolic (e.g., algebraic). Because the Analytical Engine could generate new programs as well as numbers, it opened up a vast new territory for the analysis of information.
From Note A, p. 696
The former engine [the Difference Engine] is in its nature strictly arithmetical, and the results it can arrive at lie within a very clearly defined and restricted range, while there is no finite line of demarcation which limits the powers of the Analytical Engine. These powers are co-extensive with our knowledge of the laws of analysis itself, and need be bounded only with our acquaintance with the latter. Indeed we may consider the engine as the material and mechanical representative of analysis, and that our actual working powers in this department of human study will be enabled more effectually than heretofore to keep pace with our theoretical knowledge of its principles and laws, through the complete control which the engine gives us over the executive manipulation of algebraical and numerical symbols.