by

P. H. NIDDITCH

LONDON : ROUTLEDGE & KEGAN PAUL

NEW YORK : DOVER PUBLISHING INC.

1 . Purpose and Langauge of the Book | . . . | 1 |

1.1 . Purpose of Book | . . . | 1 |

1.2 . Language of Book | . . . | 1 |

2 . Aristotle's Syllogistic | . . . | 3 |

2.1 . Mathematical Logic Produice by Four Lines of Thought | . . . | 3 |

2.2 . Aristotle's Work on Logic | . . . | 3 |

2.3 . Reasoning, Implication and Validity | . . . | 4 |

2.3 . General Statements | . . . | 5 |

2.4 . Aristotle's Syllogistic | . . . | 6 |

2.5 . The Use of Variables by Aristotle | . . . | 8 |

2.6 . Aristotle's Errors | . . . | 10 |

2.7 . The Megarian and Stoic Logic of Statement Connections | . . . | 12 |

3 . The Idea of a Complete, Automatic Language for Reasoning | . . . | 14 |

3.1 . Lull's Ars Magna' | . . . | 14 |

3.2 . Dalgarno's and Wilkin's Languages | . . . | 15 |

3.3 . Descartes' Idea of a General Language | . . . | 18 |

3.4 . Leibniz's Idea of a Mathematics of Thought | . . . | 19 |

3.5 . Leibniz's Errors | . . . | 20 |

4 . Changes in Algebra and Geometry, 1825-1900 | . . . | 23 |

4.1 . Peacock's Algebra | . . . | 23 |

4.2 . Equations of the Fifth Degree | . . . | 25 |

4.3 . Vector and Matrix Algebra | . . . | 26 |

4.4 . Geometries of Euclid and Lobachevsky | . . . | 28 |

5 . Consistency and Metamathematics | . . . | 30 |

5.1 . Mathematics and Systems of Deductions : Consistency | . . . | 30 |

5.2 . Metamathematics | . . . | 32 |

6 . Boole's Algebra of Logic | . . . | 33 |

6.1 . Boole and the older Logics of Aristotle and Hamilton - De Morgan | . . . | 33 |

6.2 . Boole the Man | . . . | 36 |

6.3 . Boole's 'Mathematical Analysis of Logic' | . . . | 37 |

6.4 . Class and statement Readings of Boole's Algebra | . . . | 38 |

6.5 . Boole's 'Laws of Thought' | . . . | 39 |

6.6 . Boole's Algebra and Number Algebra | . . . | 42 |

7 . The Algebra of Logic after Boole : Jevons, Peirce and Schroeder | . . . | 44 |

7.1 . Later Developments of Boole's Algebra | . . . | 44 |

7.2 . The Work of Jevons | . . . | 44 |

7.3 . Caomparison of Boole's and Jevons's Systems | . . . | 46 |

7.4 . Peirce's Points of Agreement with Boole and Jevons | . . . | 48 |

7.5 . Peirce's Use of the 'A Part of ' Relation | . . . | 49 |

7.6 . Peirce's Theory of Relatons | . . . | 50 |

7.7 . Peirce's Logic of Statement Connections | . . . | 52 |

7.8 . Peirce's Quantifier Logic | . . . | 55 |

7.9 . Schroeder's Algebra of Logic | . . . | 56 |

7.10. Whitehead's and Huntington's Work on the Algebra of Logic | . . . | 58 |

8 . Frege's Logic | . . . | 59 |

8.1 . The Mathematics of Logic and the Logic of Mathematics | . . . | 59 |

8.2 . Reasons Why Frege's Writings were Undervalued | . . . | 60 |

8.3 . Frege's View of the Relation between Logic and Arithmetic | . . . | 61 |

8.4 . Frege on Functions | . . . | 63 |

8.5 . Functions as Rules | . . . | 63 |

9 . Cantor's Arithmetic fo Classes | . . . | 66 |

9.1 . Finite and Enumeral Classes | . . . | 66 |

9.2 . Cantor's Cardinal Arithmetic | . . . | 68 |

9.3 . Cantor's Ordinal Arithmetic | . . . | 69 |

9.4 . Burali-Forti's and Russell's Discoveries | . . . | 69 |

10 . Peano's Logic | . . . | 73 |

10.1 . Implication and Mathematics | . . . | 73 |

10.2 . Peano's Purpose in his Logic | . . . | 74 |

10.3 . Some of Peano's Discoveries and Inventions | . . . | 75 |

10.4 . Pean's School of Metamathematics | . . . | 76 |

11 . Whitehead and Russell's 'Principia Mathematica' | . . . | 77 |

11.1 . Short account of the three parts of 'Principia Mathematica'. | . . . | 77 |

11.2 . Relation between Russell's 'Principles' and 'Principia Mathematica' | . . . | 78 |

11.3 . Russell's way of getting consistency in class theory. | . . . | 78 |

12 . Mathematical Logic after "Principia Mathematica" : Hilbert's Metamathematica | . . . | 79 |

12.1 . The Growth of Logic after 'Principia Mathematica' | . . . | 79 |

12.2 . The Structure of an Axiom System | . . . | 80 |

12.3 . Post's, Hibert and Ackermann's, and Gordel's Demonstrations for
'Principle Mathematica' Systems | . . . | 81 |

12.4 . Brouwer's Doubts and Hilbert's Metamathematics | . . . | 82 |

12.5 . Goedel's Theorem in the Metamathemaics of Arithmetic | . . . | 84 |

12.6 .The Theory of Recursive Functions | . . . | 84 |

Further Reading | . . . | 86 |

Index | . . . | 87 |

Back cover | ||

Review . |

Modern Logic, like mathematics, is a highly-specialised subject with an extensive technical literature. but unlike mathematics it is a relative newcomer to the academic scene. While it s true that the initial stages of its develoment took place several hundred years ago, it made its bebut only as recetnly as the beginning of the present century. Since then it has progressed so rapidly that few textbooks on the subject can hope to be comprehensive. A compromise between breadth of coverage and depth of treatment seems inevitable. The present series is designed to meet this problem. Each monograph devtes separate attetion to a particular branch of mdern logic, the level of treatment being intermediate between the elementary and the advanced. In this way a wider coverage of htis subject will, it is hoped, be made more acessible to a larger number of readers.

All the special words of logic which are not among the 850 words of Basic English are put in sloping print when they are first made use of here and their senses are made clear by Basic English.

However, though we have done our best, by using Basic English, to keep the account as clear and simple as possible, it will not be surprising if there are bits of it which are hard fully to get a grip of in a first reading ; this is specially true when the newer developments are being quickly talked about (9.1 to the end of the book). The reason for this is that the tendency of the growth of all sciences is towards the more and more complex and towards an ever-increasing number of high-level ideas in their organization. Certainly the growth of Mathematical Logic has been like this. But our hope is to have said nothing that will not be completely clear after some thought -- thought will sometimes be needed . No earlier knowledge of logic will in any degree be necssary and only such a knowledge of high school mathematics as the reader stil has in the mists of his memory.

Now, more generally, let s1, s2, . . ., s

It is only right to say in addition that Aristoltle gave

Reasons for the over-great respect for Aristotle's logic were at first his system's being the only one of which one had records making possible detailed knowledge, together with the authority of the Church of Rome which had taken up, with some adjustments, much of Aristotle's teaching in philosophy ; and later the crushing effect of higher eduction whlch was controlled by those who had given their years to the learning of the languages, history and writings of the Greeks and Romans and so naturally put the highest value on such knowledge, a value much higher than they were ready to give to new sciences and to new, natural knowledge.

An example of a syllogistic implication having validity is ' if all animal is going to die and all man is animal, then necessarily all man is going to die'. This implication is of the form ' if P is truly said of all M and M is truly said of all S, then necessarily P is truly said of all S'. (The reason for saying 'man' and 'animal' and not 'men' and 'animals' will be made clear later, in 6.2.)

If one keeps in mind that the Greeks were very uncertain about 'and very far from letting variables take the place of numbers or number words in algebra, which was why they made little headway in that branch of mathematics though they were good at geometry and so why the invention of a geometry based on algebra was not made till number variables had become common and were no longer strange, in the time of Fermat and Descartes, then there will be less danger of Aristotle's invention of variables for use in Syllogistic being overlooked or undervalued. Because of this idea of his, logic was sent off from the very start on the right lines. But not before Boole, 2200 years later, who saw in the old Syllogistic the seeds of an algebra of logic, was the important step taken of changing Syllogistic into a mathematics of certain reasoning. Only when logic was married to mathematics did it become fertile. On its becoming the mother of a good number of ideas and teachings of profit, logic's position as the poor relation of philosophy came to an end, the change going so far as to make the new logic widely looked at and respected as the head of the family. In fact, it has been the desire of some persons to see the rest of the family dead. This is true of the supporters of the Vienna Circle that was flowering in the 1920's and 1930's : it was their belief that logic, in the wider sense that takes into account, in addition to Mathematical Logic, the logic of chance and the logic of the natural sciences -- logics which have again, like Mathematical Logic, deep connections with mathematics -- is the one part of philosophy that is of serious interest and value, because it is the only one in which true knowledge is possible ; in the other parts man's thought is wasted on systems of fictions caused by the tricks of language and on talk which, though put forward as offering a key to the secrets of all that there is, is without good sense, by reason of its being framed in such a way that its powers of opening the locks of existence are unable to he publicly tested by experience.

Some points in the first group are these. In the old logic only two general statements as conditions and, in reasonings having validity, not more than three general names may be used in implications. But there is no good reason for so limiting implications while there is a good one for ot so doing, this being that the business of logic is with every sort of implication having validity and there are implications having validity which have more than two general statements as conditions and make use of more than three general names ; an example is ' if all animal with wings is backboned, no self-moving thing is rooted in the earth and all that is backboned is self-moving, then necessarily no animal with wings is rooted in the earth'. Again, Aristotle's logic is not a well worked out system like the systems of mathematics ; in these the different statements are or are made to be dependent on one another : one takes a selection of some of them as starting-points from which the rest have to be got by deductions. Though, it is true, Aristotle's Syllogistic is different only in degree from for example Euclid's system of geometry, still the degree between them is enough for it not to be unkind to say that Aristotle's theory is nearer to a simple grouping than to an ordering and organization of his material. In addition, its use of signs other than words

There are two chief points in the second group about which we will say something, again without going into details. Aristotle and his school of logic made no attempt to give an account of the theory of implications having structures whose validity is dependent on the force of words like ' not' , ' and', ' or' and ' only if ' which are used for forming more complex statements from given ones ; an example of such a structure is ' if s1 is false but s1 is true or s2 is true, then necessarily s2 is true', and another example is ' if s1 only if s2 but s2 is false, then necessarily s1 is false'. This branch of logic may be named the logic of statement connections. It was first given attention by the Megarians and Stoics, working quite separately from Aristotle ; see 2.8. Secondly, the old logic did not take into account implications such as ' if A is greater than B and F is greater than C, then necessarily A is greater than C', ' if A is B, then necessarily A of X is B of X'. If song is music, then necessarily the song of a bird is the music of a bird') and if A is B, then necessarily X of A is X of B' (if Christ is the Highest Being, then necessarily the mother of Christ is the mother of the Highest Being', an example given by Leibniz), which are dependent on words or ideas for relations between things. The expansion of the old logic so that at least some par of the logic of relations might be covered was first undertaken by De Morgan in the 1840's ; in his view the logic of relations was more important because wider than Syllogistic, this being nothing but a small part of that logic.

The Megarian work on logic was done between, roughly, 400 and 275 and the Stoic between, roughly, 300 and 200 before Christ. There were two chief ways in which the Megarian-Stoic theory was different from Aristotle's Firstly, it was interested in forms of reasoning having the structure of an argument and not the structure of an implication. Secondly and more importantly, it as a logic of statement and connections and not one of general and suchlike statements.

The sorts of reasoning to which the Megarians and Stoics gave attention were these; ' if s1 then s2, and s1 ; so s2', ' if s1 then s2, and not s2 ; so not s1' ; 'not the two of s1 and s2, and s1 ; so not s2' ; ' s1 or s2, and s1 ; so not ' s2' ; and ' s1 or s2, and not s1 ; so s2'.

Some of these and other forms have an ' if-then' statement as one of the bases of the argument. Because one may rightly put forward as true what comes at the end of an argument only when all the statements on which it is based are true, it is necessary for an answer to be given to the question : when is an ' if-then' statement true ? This question was much talked about in the Meganian-Stoic school, so much so that someone (Callimachus) said : 'Even the birds on the rooftops are having discussions on the question.' A number of different answers to it were offered ; we will give three of them.

. . .
### The following has not been edited !!!

Sections 5.1 to 6.2 may be okay. 6.3 and 6.4 are still bad.

Sections 6.5 and 6.6 are good.

**3.2 . Dalgarno's and Wilkin's Languages**
After
Lull suggestions for a general language for a general
science were not uncommon in philosophy. But the
development of such a language was not made till the
1660's when *Ars Signorum* (' The Art of Signs') of
George Dalgarno (1626-1687) and the *Essay towards a
Real Character and a Philosophical Langauge* of John
Wilkins (1614-1672) and were printed. The ideas at the
ase of Wilkins's *Essay* were, to some degree, copied
from Dalgarno. However. it is not by chance or without
good reason that Wilkins's book has been better
kept in memory than Dalgarno's. The working out in
detail of the ideas for a new consciously made language
is more valued than an outline of them and the Essay,
the last and separate part of which (by William Lloyd)
gives a full list of the English words of the day and
their senses in Wilkins's language, is on a much greater
scale than Dalgarno's. Things were not so hard for
Wilkins, who at one time or another was a Head of
Colleges at Oxford and Cambridge -- the only man
ever to have been so at the two places -- and who had a
high position in the Church of England ; he had the
backing of the newly formed Royal Society, of which
he was one of the Secretaries, and his book was printed
by the order and the necessary money came out of the
pocket of the Society. Dalgarno, on the other hand,
was a private school teacher.

Wilkins's purpose was, he says, to give a regular list and account of all those things and ideas on which marks or names might be put in agreement with their natural properties, is being the end and design of the different branches of science that all things and ideas be placed in such a frame as may make clear their natureal order, their ways of being dependent on one another and their relations. (It is easy to see here the - 15 - effect of Bacon's teaching on the roads and limits to knowledge in the sciences : science is the discovery and ordering of natural groups of facts, not the building of delicate theories of them.) Wilkins goes on to say that an art of natural rules for forming statements and questions is needed from which one will get ' such helps and instrument s' as are necessary for the framing of the simpler ideas into smooth and unbroken talk or writing. The smaller and simpler this body of natural rules is, the better. Wilkins had three points in mind : the rules of his language were to be 'natural" ; the 57817 :un > A n .¢-.».P' s . 4_ .!9.l?9. o n 41 I I l | I 4 : 8 .. gSSgfvwvulpg n 4- .-,--q'1'1 Wi ~ ... | l l ¢..t .:. ._ of which they I 's * *rr* 5 QI these . *"* n L 1. » . H . , so that the names would in some way be in agreement with the things they were signs of. In addition, Wilkins, like Dalgarno before him, had the opinion that a system of shorthand writing, in which the shorthand outlines are dependent only on the sounds of the words, was needed for a specially designed language such as his and no small part of his book is given to the development of this side of his language,

The great trouble Wilkins took to make his language a living instrument of science (and international trade and religion) was completely wasted-but for its helping to keep before men's minds the thought that it is possible for a language better than the everyday ones to he formed by conscious art. The offspring of Wilkins's brain was dead at birth. One reason for this was that - 16 - because of discoveries in science and the changes in common knowledge caused by these it came to be seen that his accounts of things in relation to 40 chief headings and under these headings an attempted grouping of all other things by divisions of higher and lower levels, in this way giving an account of everything by what it has in common with certain other things and how it is different from them, as in taxonomy-were false or without support, and, equally serious, ir was seen that the new ideas and sorts of facts that were the outcome of the new knowledge were not covered by Wilkins at all, while they put in the shade much of what was covered, for example about the Highest Being and other such ideas that were a necessary part of the beliefs of a man having authority in the Church.. . ¢.+ ; A a =i,,§* why W|.lkin.§'s,sg11||1:|;nn,,,l:nd. no "'¢;» "T "" .U 11. .m. . . ' 1.ii ..| \ . 1 1 natural blow . dge was very much o 'ii d by him. On condition that a language is ab to be used for making as sharp and as detailed recordings of observations as is desired and is able to undergo expansion, letting new ranges of fact and of thought be talked about without unnecessary trouble being caused by the language apparatus, then it is not generally important for the natural sciences if the Language is not as good as possible in other ways, for example in having forms that are not completely regular. On the other hand, languages, such as those of mathematics, having short and simple signs and for which there are rules for building statements which put forward two groups of numbers or of other things as being equal or being unequal and for going from some given statements of this sort to other ones, by processes of deduction, do have great powers of increasing and ordering knowledge. But this knowledge is of possible - 17- relations between things, not of facts themselves. Wilkins did not see that if a designed language was to be of use in the way he was picturing to himself, it would have to be a sort of arithmetic or algebra. However, he was not expert enough in mathematics to be able to make the necessary developments for this new purpose .

**3.3 . Descartes' Idea of a General Language**
It seems that Descartes (1596-1650) was the first person
to have the idea of a general language -- ;une langue universelle' are his words rot it -- as a sort of arithmetic.
In s a letter to Mersenne of 20 November 1629 he said
that the invention of a language is possible in which
; an order is formed between all the thoughts which
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. . .The invention of this language is dependent on the
true philosophy, but for this it is not possible
for all the thoughts of men to be listed and put in
order or even to be made separate from one another so
that they become clear and simple, which is in my
opinion the great secret of getting well-based knowledge.
And if someone was to give a good account of
which are the simple ideas that are in men's minds of
which all their thoughts are made up . . . there would
he almost no chance of going wrong' No doubt wisely
Descartes made no attempt to give a list of all our
simple thoughts and to put them in order so that an
arithmetic of reasoning might be formed that would
let one get complete and certain knowledge of whatever
is true.
- 18 -

**3.4 . Leibniz's Idea of a Mathematics of Thought**
A little later Leibniz (1646-1716) had designs for a new
and general language that were not unlike Descartes', but
he took them some stages further. From as early as his
*De Arte Combinatqria* ('On the Art of Complex Forms') ,
printed in 1666, Leibniz made suggestions fot. a
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Leibniz was ruled by a belief in mathematics as the *.scanned.toe~..v... . . .£ . #I 5 l nv: l 9; are able £9 undergo 1 . . f* "* . I .&=:t.d49no~|»»m.s11 s@»st€°n»~=sls°=a= . . . . . .4 . 4 .=| . . ntrd "m ao far as such decisions are by - 19 - reasoning from the given facts. Because though some *i expenences are ever needed as a base for reasoning, when these experiences have been given we would get from them everything that any person would ever be able to get from them, and we would even make the discovery of what more experience is necessary if our minds are to become free from the rest of our doubts. . . . If we had a body of signs that were right for the purpose of our talking about all our ideas as clearly and in as true and as detailed a way as numbers are talked about in Arithmetic or lines are talked about in the Geometry of Analysis, we would be able to do for every question, in so far as it is under the control of reasoning, that one is able to do in Arithmetic and . pendant ' and 'I of signs and by a sort of Algebra; an effect me 0 this is that the discovery of facts of great interest and attraction would become quite straightforward. It would not be necessary for our heads to be broken in bard work as much as they are now and we would certainly be able to get all the knowledge possible from the material given. In addition, we would have everyone in agreement about whatever would have been worked out , because it would be simple to have the working gone into by doing it again or by attempting tests like that of "putting out nines" in Arithmetic. And if anyone had doubts of one of my statements I would say to him : ict us do the question by using numbers, and in this way, taking pen and ink, we would quickly co me to an answer.' - 20 -

**3.5 . Leibniz's Errors**
The idea of what we have been
naming a complete and automatic language for
reasoning is made very clear by these words of Leibniz. But
though he frequently gave attention to the idea of such
- 20 -
a language he did little with it. He was at the same distance
from having a new language system at the end as
he had been in 1666. His design got no further than
being designs -- and rough ones. Leibniz went wrong in
two ways. Firstly, he was unwise not to have msfr much
.%Fef tI~fe be
desired inention.
There is no hope of covering from the very start every
part of what might be reasoned about ; the building of
any such apparatus as was pictured by Leibniz, as of
any theory in science, comes only with a slow process of
growth, not jumping suddenly into a condition of full
development at birth like Athena out of the head of
Zeus. It is necessary to go one step at a time, starting
with what is simplest; this is the best way of getting
knowledge that is of value in itself and of value as a base
for making wider the range of facts or thoughts taken
into account. Leibniz would have done much better by
limiting himself to an attempted forming of a language
for no more than one or two branches of the mathematics
of his day. Desiring to have a system that was a
guide to everything, he was unable to put together a
system that was a guide to anything. At the back of this
desire was the belief, by which Descartes earlier had
been gripped, that to a person like himself who had
made such new and important discoveries in mathematics
the invention of an instrument of logic having
the power to give the answers, without error and with-
out waste, to all that is a question for man's reason, was
a work of little trouble and of little time. This belief was
one side of the thought then current: in living memory a
great increase of knowledge had taken place and the
mind had become much more free and open, had
become forward-looking and full of hope, sharply
conscious of its powers.
- 21 -

Secondly, Leibniz went wrong by desireing to have 4 J I. I an . I '.~~ i .., \ . 1: v »=¢1r»\lln41e a : 1 :Wt ~ m<>£1°d¢.,=iru1s4§r:.:sese the .. ~. " ~~a 4 ~tem in wbish #li '=1\!f=~U1l§11* be . xit:?Wdu1d 'Be or ei-EH-a" system of . "~¥M"i>1=@==~w1>hs=. The reward would have been greater if he had made the decision to keep to the logic, which is so much more straightforward and which would have given him the chance of turning the metal of the old logic into the gold of a new and better one, for example one whose laws, unlike those of Syllogistic, are enough and have the right properties so that the validity of arguments in at least some of the theories of mathematics may be tested and judged. Or, if Leibniz had the feeling that the material in addition to the form of reasoning had to be a part of his language system, he might have done well -- much better than he did -- by letting himself be interested in the bricks of only mathematics, having as his purpose an organization of mathematics based on its simplest ideas and starting- points. A road to the full development of such an organization was not seen till about 200 years later : by Frege, Peano, Whitehead and Russell, in the years between 1875 and 1910.

It would not be right to get the idea that Leibniz's work was of no value at all because his controlling thought of a new mathematics-like language came to nothing. Having that thought, it was natural that his mind was transported in the direction of logic itself and there he made a number of important discoveries some of them in Syllogistic but most of them in fields of 1logic that were new or had been given little attention befpre. However, these discoveries were not taken seriously till much later because the separate bits were

- 22 -

not united in a general theory and so, not being complete enough, they were not put in print or offered to the public in Leibniz's time or even till long after; by then they were common knowledge, not as his teachings but as the teachings of theories worked out in the years after 1840, theories whose start and development were in no way dependent on Leibniz's work on logic.### 4 4 . CHANGES IN ALGEGRA AND GEOMETRY, 1825-1900

**4.1 . Peacock's Algebra**
Between 1825 and 1900
algebra and geometry underwent great changes, changes
producing by 1900 a completely different general outlook
in the philosophy of mathematics from that which
had been common before. The changed forms and
purposes of algebra and geometry had a strong effect on
all the parts and at every stage in the growth of
Mathematical Logic.

An equation is a statement that two things, normally two groups of number. or of signs representative of numbers, are equal. Till 1825 or a little after that, .=.=.=.~.:w... 4§s¢ I • o I ll >... ». ;.¢ ¢.....,»..;4 *.#&'4l»81%;*9l¢.9x.7<:. -q ,re Q , . . ..4- ¢... .7."= "*°~ . " .. ...|n 1: .. . -A | x ~i;"""l *%~8'¢¥iiW .,. Fda learning about in high school, for example 3x-2y = 11 - 23 - and ax'+bx+c =0. The business of the theory was to get a knowledge of how such equations may be worked out to give number values which make them true and M get a knowledge of the conditions controlling the existence of and relations between the number values. The four operations of addition and so on were dOne, as they are still done by schoolboys, more or less un- consciously by making whatever moves seemed right and natural, the rules supporting these moves resting in the dark. There was no thought that a statement of the rules was necessary or might be a help in the development of algebra. It is strange that though geometry had early in its history been turned into a structure of fixed starting-points and of reasonings from these, the position of algebra, like arithmetic, was dilferent. The need of forming a table of and of being consciously guided by the laws of algebra which say what are the properties of the operations in algebra was seen only by Peacock (1791-1858). In his book*A
Treatise on Algebra *(l830; second writing, in two parts,
1842-1845) the idea was put forward that algebra is,
rightly viewed, a science of deductions like geometry.
(At about the same time -1837- the first account of
Newton's mechanics as a science of deductions, copying
Euclid's Elements, was given by Whewell, a friend of
Peacock, in his Mechanical Euclid.) Peacock had two
chief points which are to be noted here. Firstly, all the
processes of algebra have to l.;e...based _on acomolete
stat nt t h e b o d of 13y§-whiQ138£e_about the I
operations used in those processes, no pro erty of an
operation_b__ejg,g,_3;§_eg.l.j t e ~~ Examples of the laws
.g1 9 i * . _of 419 4 is |
af . £ t l. . .. - . 44 . 14 j ' . . Qi L ..;
I ti i ?» n outcoméis || . " I i
. . . 48 ` . ... gg.8 7 " . . .. . u 4
U 11 r + vanabfes is changed: a+b .
. .1" . . `u8.4 1"* ; §. ! F . . t " *d9 ii . . . . rmreiagg ¢ ., 1
n| ! .: ¢jat _in outcome is l g . 4 . tile g. ;.;. H. .. 4 . 4. . 4 3;
IF e . w .* 4$ > * se . . !r.;.
. . * *..wa . v . y ¢*|i4--*--»-
w5. r§Q"1 * ii . a . . . . .. . . .
. t r H* . . . . .r 4: . .3r§§ I1* .. . . . 1. """"3* l r sae I 110 I 1. n I
av ` r 5=ax c. P
-25-
~~

**4.2 . Equations of the Fifth Degree**
Another event
causing the movement in the direction of Abstract
Algebra and away from the limits of the old theory of
equations was the surprising discovery br Abel and
others n numlbtt .
..w'§»'8ll»ll¢%istA
..9..H.! ."=:i°°s. ..¢t..s8¢?§!"e3s...4:
. 1 . . . .. . : I 0
.£91 : . m. . ..9r?" we * _8 1 . .. 5..$ 8* . . . F; |1 .
.\. |" | be y can Nzetie- H prooenews
- 25 -
specially the taking of square and higher roots-which
make possible the automatic getting of number values
for equations of degree 1, 2, 3 or 4. The belief had
become fixed that answers to questions about the
number values for equations of higher degree than the
fourth would be (and anyhow that it would not be
possible to make certain that they may not be) got
in the same simple way as for those equations of no
higher than the fourth degree. Abel's discovery was
seriously damaging to the belief, having a wide distribution
among workers in science and philosophy, that at
least the heart of future knowledge will be in agreement
with the expert knowledge of their time. This
belief had to take blow after blow, not only in mathematics
but in the natural sciences, through the years
after this discovery in algebra, some of the blows
coming from Mathematical Logic.

Galois, in attempting t ; |;.|. 1 ' ' ` to Nwmwagntai _ I I F =;¢&4; -.. : nm5 a u *"§¥;2.Zi;3v ~;. I ... &. > < 5 4 o I » ri°"*!§~ . . . m. >. .. su : . u -~ . .. 4: * 8"1'l1|* / si. . . + . ..p¢=¢x »£ e " 1 . ku quickly into being as the first new branch of Abstract Algebra. In this way the changes in algebra taking place in England came together in harmony with those taking place in other countries of Europe.

**4.3 . Vector and Matrix Algebra**
To those who were
pained by the discoveries of Abel and Galois, worse
was to come. They were given more, violent shocks by
the systems of Vector Algebra and Matrix Algebra
worked out not long after the death of Galois in 1812.
These systems were company for Group Theory as new
branches of Abstract Algebra.
- 26 -
' I U i 4 . ... . * *lux MIL jr . 41.s . ur . n 1 . . . . .. ... ..\. - " . " " ...n4 ;i ..3.. 5W *§&iv .. ¢... ,.§;¢ 4;:5 4|lt=\r
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..m I 6 . . i IL l n . . .. . .x
4*1'¢W'*t...|4|-¢Hin0. .4 I • gr ¢ - P in a p 1 e . i .. .*.. 1 § . .
W ER. 8.. -.
L *vw IM. tt: 3 . w1.-= *W* Lyw.8, H»=m1=m4 .. 1 . *I l 1 + i 2 I | . . * the J* . §. . . . . . .¢.; . M gg.. . . 4 *is s.:
li3I .4I4h.l|9.| I tally .+ tu . . ... .. '*= " . .... . . . . . 89%... . 8 .. .. :rams- .
J* r ways like l Ol .._ .
.5 . . .=¢,@.. .: . 1 . a. . n . . 4 | 1.
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1 . ,1ggh|4um
18 that ax c)-=(axh}+(axc). (Signs for
direction-properties and for operations on them are
normally put in blacker print.)

The 1850's saw I . * . 1 ..i . t 85 dn§¥a,; 4 il •. k-. .. . Q; &1.*2 = 1 | | a. $...i . I. 0 . . . 088.831 .'>. . th»:.sa.n1¢_m1he. 9 .9 . • . . ;" 4.3 . 4 §| . " . . 0. . . • . . . . . . 1 1 . . as, a . . .. ... . • _ *.. .i . . . .. . I . r .r . . . . . . . s . 4. i pa - .. '. . • . ". I .E . an .. . Algebra M lhwi wr 8"|3§BxA is not generally true. To take a simple 3 example: let A=(l 2) and B=( ) ;then by the sense 4 given to :=< in connection with number tables AxB - (1 =(1.3+2.4]=ll1), while on the other hand (3) .. 6• BxA (1 2)=(3 I his ¢|,¢"¢h;ttl1ue.¢ _ (8) 4 8) U .. . . . 2 . . . s . . . . . . -a. ... ... . \. .. .S|{..1. ¢...M |e-:1=1e..|: . *4 l. * .... .8. T 4 . JE'.j .4.. .. .." .. . a e .. ss .v . . ... 2. . _ " v . . . ;L. .: ; .l I rifihw hnmdmummnmMxnMHm»umm4d9 - 27 - . 1 . F g,. ...- 4 .4 . ni t. . . . r . . 8 9. * . :. . .. 1 " ' *gag " 1 ..` 4 = . ". H. - " |4 . i =M=ti11s pount 8 ,rr V74 Yu I hr | any * > . .. .* .r¢~ . r.. * " " . . .I . 1: L ..* . . 5 +e!? | . .. *984. . vi ". . 9 : . .. . " . x 4 . : T r - an-pnI=@in1||»\un:d33i¢ • * .- 4 f . .. . . .. . 4 .. . In ; . J 9. . 4. . . . . . . . 81lM!\lih¢¢2n1§»cQnsd0W H P ra - 4 a. *rcllsar what mm 3h¢*-3¢ilF§;l§gg * : . . . . . . ..4.= ... . a.. used in the - ..c .. ;1i&¢rnnt. Belds . . 4 . . 8 . a.. . 1 M||gt9|=Q8 f . . "aa . . 41 .. . 1Im. 2 .. ;2 \ . . 2 . e mane i. M*r i | | Qn¢, . . This £1138586 .in algebra, whose full . >¢» ==a~r. . . . "Q": * .. . ... . . . . ). vwna Qigxeat. wish*. . 4 . v in the development of almost every other part of mathematics. To give only one example of the way in w c Mathematical Logic was touched by it: because of the new current of ideas Schroeder and Whitehead in the 1890's were caused to give an account of the algebra of logic in the form of a science of deductions using to no small degree the signs and processes of algebra and of its latest directions in Abstract Algebra. These workers, in their turn, were copied by others, and it became the normal thing in Mathematical Logic to take note of the latest theories in algebra, by which it is offered new food for thought that keeps it a living branch of philosophy and science.

- 28 -

**4.4 . Geometries of Euclid and Lobachevsky**
1
as Geometries of Euclid and Lobachevsky. Fo6
. . *
. . .. I ,s\=rs°s=1h¢4~*e*¢* . . . . U r... .
. .A I I ng; oin8~ Thi Ent in! . . . . ik := . 4 time ..
. . . . .* .
* .. . .
» ua be true without the support of reamonmg.
y 's doing this, geometry was changed and kept from
being nothing but a mass of separate statements and

- 28 - was turned into a united body of knowledge, all the parts of which were dependent and in connection with one another. Because of Euclid's work geometry was_ looked at down the years as the most complete branch of mathematics and its deduction form was looked at as offering the best way to true knowledge and, outside religon, the highest knowledge.

--~» -~'~t was thc invention by.»||...~.,:...... it | | e ml. J 4 33 \ ¢|§=|.l¢a»*q ~"s...~gp..n..»34 a~...~ ' .. 4.. 'L' I 1 .»» glggqg8 4. ...»4.*...~w.,_»._.~~;..~1.~ 1 =~>=~1»~ 1 4-~ ` e -. s More than one l I | ' 1 parallel to the given line. Other ways in which they are different are outcomes of this; for example the law of Pythagoras is true and may be got by deduction in Euclid's but is false and may not be got by deduction in Lobachevsky's geometry. The fact that one was able to have a system other than Euclid's came as a deep shock, producing in a quite short time a view of what mathematics is about, what its purpose is and what its relations to physical space and things in that space are which was completely unlike the normal, old view in the philosophy of mathematics. When it was seen that one may have two geometries where what is true in one is false in the other, the old opinions such as Plato's, the Schoolmen's, Spinoza's, Leibniz's and Kant's by which mathematics is a science giving completely certain knowledge of existence -- natural or higher-than- natural, and if not of things-in-themselves, then of things as they seem to be -- were not able to be put forward with reason on their side.

The effect of the changed outlook on mathematics and specially geometry was to make theories in mathematics viewed not as in themselves natural knowledge

- 29 - sw ~. .r2:w:ij. iriir .,.4;s' -be -ia.¢|--4-4--4 a hj hlneauU|unineh I . » aiiml iinteflaqugahelpinilnmmmilacieneu. ¢1 . • llldnemnmes I due nnennpm to get lllolutniunnbersandspa1eesandsucl:|like\bings,tonuake inmueatixuug dileovelies about their properties and rehtium, and to put the supported by these dis- ouveries into systems of reasoning. Mathematics is the hee invention of the mind, limited only by possible uses in natural science that may be desired and by rules of logic that one is willing or is forced, as a person of good sense, to be guided by.

The third turning-point in the history of geometry is best talked about in connection with what comes under the fourth and last of the headings given in 2.1, so we will now go on to that.- 30 -
### 5 CONSISTENCY AND METAMATHEMATICS

**5.1 . Mathematics and Systems of Deductions : Consistency**
The idea of the parts of mathematics as
systems of deductions goes back to Euclid's work in
geometry, geometry being the first part of mathematics
that was made into a system of this sort. What a system
Of deductions is has been outlined clearly enough to
the reader for us not to have to go into more detail

- 30 - about it at this stage, though more will be said later. What we will have to say something about here is another side or level of a system of deductions, the idea of which has been very importance for Mathematical Logic.

Some attention in relation to geometry had been given before the 1890's to the question : Is it possible to get the end of one chain of deduction in a system a statement S, which becomes by this fact a theorem -- a true statement -- of the system and to get at the end of another such chain the theorem not-s which says that s is after all, false ? A system of deductions is said to have the property of*consistency* if and only if it does not
have two opposite theorems s and not-s. The amount
of attention that had been given to the question of
consistency had been small because the question had
not been taken very seriously, the belief being general
in mathematics that any expert was able quickly to see
the answer to the question of a system's consistency
without his having or needing a supporting argument.
But in the 1890's the question was viewed as interesting
and important by that Italian school of mathematics
that was headed by Peano. From the point of view of
logic there is no doubt the question is an important
one because any statement whatever abut the field
under discussion becomes a theorem in a system that
does not have the property of consistency, the statements
being true ones or being false ones about the
field, so htat what is true would not be able to be kept
separate from what is false, while the very purpose of a
system is to have true nad only true statements about
the field as theorems. Because the question of consistency
is important, how the answer is made and supported
is important. Peano's opinion was that reasoning
and clear public tests are the right instruments for
making decisions about consistency. (See end of
9.4.) - 31 -

He had the same opinion about another question that might be put about a system of deductions. Giving the name*axiom* to a statement that is taken to be true,
without the support of reasoning, at the base of such a
system, this question is : Are an of the axioms of the
system dependent on the others in the sense that it is
possible to get an axiom as a theorem at the end of a
deduction in which the axiom it self is not used?

Some thought had been given to this question and the question of consistency from quite early times in connection with attempts to get Euclid's axiom of parallels as a step in a deduction not resting on that axiom, these attempts being made to let one see that that axiom is dependent on the other axioms : which it is not. However, only in the 1890's were these questions regularly put, and put about all systems of deductions in mathematics, and were ways designed for answering them by arguments. Peano and his school were chiefly. responsible for this, though from late in the 1890's the work, in the same direction, that was done by Hilbert in Germany had a greater effect on later developments.

**5.2. Metamathematics.** From this work came a new
branch of Mathematical Logic : metamathematics. The
business of metamathematics is with what may be truly
said about some system ( or group of systems) of deductions,
for example that it has the property of consistency
or that no axiom is dependent on the others. If S is a
system whose field under discussion is F, then the purpose
of S is the building of a framework of supportable
theorems about F, the theorems saying what true relations
there are between the things in F. If S is an
arithmetic, then a theorem in S such as ' 7 + 5= -12 ' is
- 32 -
about numbers. On the other hand, ' "'7 + 5 = 12 " is a
theorem in S; ' is not about numbers ; it is a statement
about a statement about numbers. ' 7 + 5 = 12 ' is a
theorem in S ; ' "7 + 5 = 12 " is a theorem in S' is not a
theorem in S but a theorem in the metamathematics S'
of S. It is necessary for the reader to see what this delicate
point is that makes S and S' different from one another.
Every system S has its metamathematics S',
sometimes named its ' logic'\ ', which is formed of the
true statement, and the argueements for them, that may
be made about about S, while S itself is formed of the true
statements and the arguments for them be
made about F ; for example, if SF is an arithmetic of the
common so , then F is the body of all the numbers
such as 34, 1/2, -6/79. S is normally a part of mathematics,
though systems of deductions outside mathematics
are possible ; S' is a part of logic: the logic of
that part S of mathematics. The birth of Mathematical
Logic was in the 1840's. Most of its more important
and more interesting developments after its first 50
years of growth have been at the level of metamathematics.
- 33 -
### 6 BOOLE'S ALGEBRA OF LOGIC

**6.1 . Boole and the older Logics of Aristotle and Hamilton - De Morgan**
Now that the four currents of
thought named in 2.1 have been talked about, we may
have a look at that stretch of Mathematical Logic which
their motion and driving-force have so far produced ;

as one might say, the look will be given from the footway at the side of the river and not from the point of view of one having a swim underwater, because clearly we are not able to go deeply into Mathematical Logic here.

What was the start of Mathematical Logic ? The shortest and simplest answer is George Boole's*Mathematical Analysis of Logic* (full name: *The Mathematical
Analysis of Logic, being an essay towards a calculus of
of deductive reasoning*) There is nothing
completely new under the sun. Every birth is the
outcome of earlier events. So Boole's little book of 82
pages was only marking a stage in an unbroken line of
thought from the past. However, though ir certainly
had connections with what some others had done, it
was still different enough from the rest for ir rightly to
be seen as starting a quite new theory -- the theory of
Mathematical Logic -- and not as being another step in
an old one .

The earlier teachings in logic of which Boole had a knowledge and which had an effect on him were, on the one hand, those of Sir William Hamilton (1788-1856) and De Morgan (1806-1871) on the theory that was based on the changing of the four, A, E, I, and O forms into a greater number of forms in which the amount of the predicate is given ; for example, Hamilton has two A forms, one being ' all S is all P ' and the other being ' all S is some P '. For later developments two points were important in the theory of Hamilton and De Morgan (the two of whom had a bitter fight, Hamilton protesting that his ideas had been taken from him without his being given credit for them, and De Morgan answering that his account was not at all dependent on anything that Hamilton had ever said). One point - 34 - was the weight placed by it on the amounts of predicates in statements the old logic noting only the amounts of subjects. The value of this was that every statement of the subject-predicate form was able to be turned into an equation or into a statement that an equation was false. By placing this weight on the idea of an equation, logic was moved near to algebra. Secondly, what had been named by Aristotle the two ' ends ', S and P of a statement had generally been viewed as signs. of qualities. In the new theory of Hamilton and De Morgan S and P were changed into being signs of the things themselves that have the qualities. An example of an A statement in the old logic is ' all leaf is green ', this having the sense that the quality of green is a 'part' of the quality of leaf. From the point of view of the new theory this statement had the sense of saying that whatever thing is a leaf has the property of being green: 'all leaves are green things'; or, putting it another way, 'the group of all the things that are leaves is a "part" of all the things that are green'. Putting the change shortly, it was a change from 'all S is P' to 'all S's are P's'. Though the reading of 'all S is P' as 'all S's are P's' had sometimes taken place before Hamilton and De Morgan (for example in Whately's Logak), the reasons for so doing, when there were any conscious reasons, were not reasons of logic. And the greater minds, such as Leibniz's, had kept to the S and P form, not making use of the S's and P's form. It was because they kept to the S and P form that they were stopped from building a better logic. The better logic they had hopes of building was to be a logic of mathematics, even if for more than of mathematics, and it was to be one based on mathematics. However, reasoning in mathematics is generally not about qualities as such but about the things that have the qualities. - 35 - For example, geometry is not a theory of the qualities ' being a point ', ' being a line ' and ' being a plane ', it is a theory of the things that are points, lines and planes .

The name used in logic and mathematics for a group of all the things that have a certain simple or complex property is*class* and the things that have the propertyr
are said to be the *elements* of the class. The ideas of
class and class elements are root ideas in all present
day mathematics. The outcome of the Hamilton-De
Morgan theory was to make possible a view of logic as
being, at least in one of its branches, an algebra of classes.
Boole was the first man to have this view clearly.

**6.2 . Boole the Man**
Boole (1815-1864) was, till he
went across the sea to Ireland as the first Professor of
Mathematics at Queen's College, Cork, when this was
started in 1849, an English private schoolteacher, whose
schooling had come to an end when he was only a boy
and who had at no time any sort of higher education.
His father was in trade on a very small scale, as a shoe-maker,
and is family had little money. The older
Boole was a man of regular behaviour, living with care,
and having a love of learning and a deep interest in the
sciences. These qualities were handed down to his son,
who got from his father, in addition, a knowledge of
the simpler parts of mathematics. George Boole beaume
a teacher for his living when he was only 16 years
old. By teaching himself from books in his free time
hom teaching others, he quickly became a person of
wide learning, specially expert in mathematics. When
he was 24 he was writing papers on new mathematics
for the *Cambridge Mathematical Journal* and five years
later the Royal Society gave him a special mark of
reward for the very first paper he sent it. He is a good
example of what might be done by self-help.
-36-

**6.3 . Boole's 'Mathematical Analysis of Logic'**
Boole's first of four books was *The Mathematical
Analysis of Logic,* whose writing took only some weeks
in the spring of 1847, when he was 31. This is a work
offering a logic bsed on the mathematics, chiefly algebra ;
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outcome of them from their equations -- if they have any
necessary outcome.

But, and this was a great step forward, Boole gave in addition an account of the logic of statement connections (2.7) as an algebra ; unlike Leibniz, Hamilton and De Morgan, he saw that this part of logic was important, aland where others had been completely at a loss he was able to give a good theory of it. It is interesting that his theory was the very same in form as that of the algebra of classes. Boole was the first man to give a united theory of logic.

We have said that Boole 's algebra of statement connections is the very same in form as his algebra of classes. This is true of the body of the book. But after this had been printed an addition of some pages to make clear a number of points in it was put in, and on the very last page Boole made a note of the fact that there is one way in which the two parts of logic are different. The algebra of statement connections has all the laws of algebra of classes and it has one more law in addition.- 38 -

**6.4 . Class and statement readings of Boole's Algebra**
At the start of his opening discussion in *The
Mathematical Analysis of Logic* Boole says that those
who are in touch with the present condition of ' theory
of Symbolical Algebra ' are conscious of the fact
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third to that of a question of dynamics or optics.
.

Boole's algebra of logic is one theory; there are (at least) two systems of readings of it, one in connection with material about classes, the other in connection 4 with material about statements. For example, in the l * v 4 the 2 * 11; i . " -. L *. ... . wisest . 'I . i . .1ii.1{ i M . 4 . . .......Q£_IW 1 .. .. 55 i sa Ur .La f 1 . - i.tun d aliout,th¢8¢ .:. 8 . * $ : ... - ... \ . . . T |"* . | . 1 . .l ..4 ¢ . . . Of . .- froln MI; . . . . being the class having nothing which is an element of - 38 - as ag clenmllla5.-l-y il 'i . : . .. one e]|emenl8 ai°e . n. ; . . * " - | i$ 4W£ A Q ;I 4 l e. e su":. " saw-is .A s. .... .. .. . gg §' \ . ¢4 . .. . i..4 . * .. . m . . u N 4 ts . . 1 . . . .. ? " I 41 1 "l""*"*° \. g * i s y'8 . I * . . . "?* utnat c ¥1 * 8 " *u " . ; l*§ 9 . . . . . . . ; . . E . .I . wgoae e M . 4 93 4 :4 =h=4m1> PJ .. v J.£ ~ 4. I . g.. I 8? . * .;.}- a;||, . e t ul o *.. J, and .if 5 .. .. 34,4 1HMi!l» | . . 4 ... ss...ni*.y..".in.. .. .3 under 4, . . " . . . . . . nl .. . d' ... sign o is. The-Eil&4B4|Ehl=&|Bl|I%g& Q 4 . ."°. .. . 1 . ,wmnh . iS . H44iHm=t@i¢f4!lFIW a . 1 4 onJr-48.91119., butnoQ~th»§wo of them . . . a I *»$a?§lMl'*¥&ui#x "' . 1 *

For Boole a statement 'x' is separate form the statement. ' "x" is true ' in which 'x' is put forward as being true ; the second statement is separate from the first if only because the first might be put forward as being false and not as being true. In Boole's algebras of statement connections the statement ' "x" is true ' becomes the equation x = 1, while the statement ' "x" is false ' becomes the equation x = 0. Because every x is true or false, a law of this algebra is: x = 0 or x = 1. This law is specific to the algebra of statement connections ( see end of 6.4).- 39 -

Sections 5.1 to 6.2 may be okay. 6.3 and 6.4 are still bad.

Sections 6.5 and 6.6 are good.

Wilkins's purpose was, he says, to give a regular list and account of all those things and ideas on which marks or names might be put in agreement with their natural properties, is being the end and design of the different branches of science that all things and ideas be placed in such a frame as may make clear their natureal order, their ways of being dependent on one another and their relations. (It is easy to see here the - 15 - effect of Bacon's teaching on the roads and limits to knowledge in the sciences : science is the discovery and ordering of natural groups of facts, not the building of delicate theories of them.) Wilkins goes on to say that an art of natural rules for forming statements and questions is needed from which one will get ' such helps and instrument s' as are necessary for the framing of the simpler ideas into smooth and unbroken talk or writing. The smaller and simpler this body of natural rules is, the better. Wilkins had three points in mind : the rules of his language were to be 'natural" ; the 57817 :un > A n .¢-.».P' s . 4_ .!9.l?9. o n 41 I I l | I 4 : 8 .. gSSgfvwvulpg n 4- .-,--q'1'1 Wi ~ ... | l l ¢..t .:. ._ of which they I 's * *rr* 5 QI these . *"* n L 1. » . H . , so that the names would in some way be in agreement with the things they were signs of. In addition, Wilkins, like Dalgarno before him, had the opinion that a system of shorthand writing, in which the shorthand outlines are dependent only on the sounds of the words, was needed for a specially designed language such as his and no small part of his book is given to the development of this side of his language,

The great trouble Wilkins took to make his language a living instrument of science (and international trade and religion) was completely wasted-but for its helping to keep before men's minds the thought that it is possible for a language better than the everyday ones to he formed by conscious art. The offspring of Wilkins's brain was dead at birth. One reason for this was that - 16 - because of discoveries in science and the changes in common knowledge caused by these it came to be seen that his accounts of things in relation to 40 chief headings and under these headings an attempted grouping of all other things by divisions of higher and lower levels, in this way giving an account of everything by what it has in common with certain other things and how it is different from them, as in taxonomy-were false or without support, and, equally serious, ir was seen that the new ideas and sorts of facts that were the outcome of the new knowledge were not covered by Wilkins at all, while they put in the shade much of what was covered, for example about the Highest Being and other such ideas that were a necessary part of the beliefs of a man having authority in the Church.. . ¢.+ ; A a =i,,§* why W|.lkin.§'s,sg11||1:|;nn,,,l:nd. no "'¢;» "T "" .U 11. .m. . . ' 1.ii ..| \ . 1 1 natural blow . dge was very much o 'ii d by him. On condition that a language is ab to be used for making as sharp and as detailed recordings of observations as is desired and is able to undergo expansion, letting new ranges of fact and of thought be talked about without unnecessary trouble being caused by the language apparatus, then it is not generally important for the natural sciences if the Language is not as good as possible in other ways, for example in having forms that are not completely regular. On the other hand, languages, such as those of mathematics, having short and simple signs and for which there are rules for building statements which put forward two groups of numbers or of other things as being equal or being unequal and for going from some given statements of this sort to other ones, by processes of deduction, do have great powers of increasing and ordering knowledge. But this knowledge is of possible - 17- relations between things, not of facts themselves. Wilkins did not see that if a designed language was to be of use in the way he was picturing to himself, it would have to be a sort of arithmetic or algebra. However, he was not expert enough in mathematics to be able to make the necessary developments for this new purpose .

Leibniz was ruled by a belief in mathematics as the *.scanned.toe~..v... . . .£ . #I 5 l nv: l 9; are able £9 undergo 1 . . f* "* . I .&=:t.d49no~|»»m.s11 s@»st€°n»~=sls°=a= . . . . . .4 . 4 .=| . . ntrd "m ao far as such decisions are by - 19 - reasoning from the given facts. Because though some *i expenences are ever needed as a base for reasoning, when these experiences have been given we would get from them everything that any person would ever be able to get from them, and we would even make the discovery of what more experience is necessary if our minds are to become free from the rest of our doubts. . . . If we had a body of signs that were right for the purpose of our talking about all our ideas as clearly and in as true and as detailed a way as numbers are talked about in Arithmetic or lines are talked about in the Geometry of Analysis, we would be able to do for every question, in so far as it is under the control of reasoning, that one is able to do in Arithmetic and . pendant ' and 'I of signs and by a sort of Algebra; an effect me 0 this is that the discovery of facts of great interest and attraction would become quite straightforward. It would not be necessary for our heads to be broken in bard work as much as they are now and we would certainly be able to get all the knowledge possible from the material given. In addition, we would have everyone in agreement about whatever would have been worked out , because it would be simple to have the working gone into by doing it again or by attempting tests like that of "putting out nines" in Arithmetic. And if anyone had doubts of one of my statements I would say to him : ict us do the question by using numbers, and in this way, taking pen and ink, we would quickly co me to an answer.' - 20 -

Secondly, Leibniz went wrong by desireing to have 4 J I. I an . I '.~~ i .., \ . 1: v »=¢1r»\lln41e a : 1 :Wt ~ m<>£1°d¢.,=iru1s4§r:.:sese the .. ~. " ~~a 4 ~tem in wbish #li '=1\!f=~U1l§11* be . xit:?Wdu1d 'Be or ei-EH-a" system of . "~¥M"i>1=@==~w1>hs=. The reward would have been greater if he had made the decision to keep to the logic, which is so much more straightforward and which would have given him the chance of turning the metal of the old logic into the gold of a new and better one, for example one whose laws, unlike those of Syllogistic, are enough and have the right properties so that the validity of arguments in at least some of the theories of mathematics may be tested and judged. Or, if Leibniz had the feeling that the material in addition to the form of reasoning had to be a part of his language system, he might have done well -- much better than he did -- by letting himself be interested in the bricks of only mathematics, having as his purpose an organization of mathematics based on its simplest ideas and starting- points. A road to the full development of such an organization was not seen till about 200 years later : by Frege, Peano, Whitehead and Russell, in the years between 1875 and 1910.

It would not be right to get the idea that Leibniz's work was of no value at all because his controlling thought of a new mathematics-like language came to nothing. Having that thought, it was natural that his mind was transported in the direction of logic itself and there he made a number of important discoveries some of them in Syllogistic but most of them in fields of 1logic that were new or had been given little attention befpre. However, these discoveries were not taken seriously till much later because the separate bits were

- 22 -

not united in a general theory and so, not being complete enough, they were not put in print or offered to the public in Leibniz's time or even till long after; by then they were common knowledge, not as his teachings but as the teachings of theories worked out in the years after 1840, theories whose start and development were in no way dependent on Leibniz's work on logic.

An equation is a statement that two things, normally two groups of number. or of signs representative of numbers, are equal. Till 1825 or a little after that, .=.=.=.~.:w... 4§s¢ I • o I ll >... ». ;.¢ ¢.....,»..;4 *.#&'4l»81%;*9l¢.9x.7<:. -q ,re Q , . . ..4- ¢... .7."= "*°~ . " .. ...|n 1: .. . -A | x ~i;"""l *%~8'¢¥iiW .,. Fda learning about in high school, for example 3x-2y = 11 - 23 - and ax'+bx+c =0. The business of the theory was to get a knowledge of how such equations may be worked out to give number values which make them true and M get a knowledge of the conditions controlling the existence of and relations between the number values. The four operations of addition and so on were dOne, as they are still done by schoolboys, more or less un- consciously by making whatever moves seemed right and natural, the rules supporting these moves resting in the dark. There was no thought that a statement of the rules was necessary or might be a help in the development of algebra. It is strange that though geometry had early in its history been turned into a structure of fixed starting-points and of reasonings from these, the position of algebra, like arithmetic, was dilferent. The need of forming a table of and of being consciously guided by the laws of algebra which say what are the properties of the operations in algebra was seen only by Peacock (1791-1858). In his book

Galois, in attempting t ; |;.|. 1 ' ' ` to Nwmwagntai _ I I F =;¢&4; -.. : nm5 a u *"§¥;2.Zi;3v ~;. I ... &. > < 5 4 o I » ri°"*!§~ . . . m. >. .. su : . u -~ . .. 4: * 8"1'l1|* / si. . . + . ..p¢=¢x »£ e " 1 . ku quickly into being as the first new branch of Abstract Algebra. In this way the changes in algebra taking place in England came together in harmony with those taking place in other countries of Europe.

The 1850's saw I . * . 1 ..i . t 85 dn§¥a,; 4 il •. k-. .. . Q; &1.*2 = 1 | | a. $...i . I. 0 . . . 088.831 .'>. . th»:.sa.n1¢_m1he. 9 .9 . • . . ;" 4.3 . 4 §| . " . . 0. . . • . . . . . . 1 1 . . as, a . . .. ... . • _ *.. .i . . . .. . I . r .r . . . . . . . s . 4. i pa - .. '. . • . ". I .E . an .. . Algebra M lhwi wr 8"|3§BxA is not generally true. To take a simple 3 example: let A=(l 2) and B=( ) ;then by the sense 4 given to :=< in connection with number tables AxB - (1 =(1.3+2.4]=ll1), while on the other hand (3) .. 6• BxA (1 2)=(3 I his ¢|,¢"¢h;ttl1ue.¢ _ (8) 4 8) U .. . . . 2 . . . s . . . . . . -a. ... ... . \. .. .S|{..1. ¢...M |e-:1=1e..|: . *4 l. * .... .8. T 4 . JE'.j .4.. .. .." .. . a e .. ss .v . . ... 2. . _ " v . . . ;L. .: ; .l I rifihw hnmdmummnmMxnMHm»umm4d9 - 27 - . 1 . F g,. ...- 4 .4 . ni t. . . . r . . 8 9. * . :. . .. 1 " ' *gag " 1 ..` 4 = . ". H. - " |4 . i =M=ti11s pount 8 ,rr V74 Yu I hr | any * > . .. .* .r¢~ . r.. * " " . . .I . 1: L ..* . . 5 +e!? | . .. *984. . vi ". . 9 : . .. . " . x 4 . : T r - an-pnI=@in1||»\un:d33i¢ • * .- 4 f . .. . . .. . 4 .. . In ; . J 9. . 4. . . . . . . . 81lM!\lih¢¢2n1§»cQnsd0W H P ra - 4 a. *rcllsar what mm 3h¢*-3¢ilF§;l§gg * : . . . . . . ..4.= ... . a.. used in the - ..c .. ;1i&¢rnnt. Belds . . 4 . . 8 . a.. . 1 M||gt9|=Q8 f . . "aa . . 41 .. . 1Im. 2 .. ;2 \ . . 2 . e mane i. M*r i | | Qn¢, . . This £1138586 .in algebra, whose full . >¢» ==a~r. . . . "Q": * .. . ... . . . . ). vwna Qigxeat. wish*. . 4 . v in the development of almost every other part of mathematics. To give only one example of the way in w c Mathematical Logic was touched by it: because of the new current of ideas Schroeder and Whitehead in the 1890's were caused to give an account of the algebra of logic in the form of a science of deductions using to no small degree the signs and processes of algebra and of its latest directions in Abstract Algebra. These workers, in their turn, were copied by others, and it became the normal thing in Mathematical Logic to take note of the latest theories in algebra, by which it is offered new food for thought that keeps it a living branch of philosophy and science.

- 28 -

- 28 - was turned into a united body of knowledge, all the parts of which were dependent and in connection with one another. Because of Euclid's work geometry was_ looked at down the years as the most complete branch of mathematics and its deduction form was looked at as offering the best way to true knowledge and, outside religon, the highest knowledge.

--~» -~'~t was thc invention by.»||...~.,:...... it | | e ml. J 4 33 \ ¢|§=|.l¢a»*q ~"s...~gp..n..»34 a~...~ ' .. 4.. 'L' I 1 .»» glggqg8 4. ...»4.*...~w.,_»._.~~;..~1.~ 1 =~>=~1»~ 1 4-~ ` e -. s More than one l I | ' 1 parallel to the given line. Other ways in which they are different are outcomes of this; for example the law of Pythagoras is true and may be got by deduction in Euclid's but is false and may not be got by deduction in Lobachevsky's geometry. The fact that one was able to have a system other than Euclid's came as a deep shock, producing in a quite short time a view of what mathematics is about, what its purpose is and what its relations to physical space and things in that space are which was completely unlike the normal, old view in the philosophy of mathematics. When it was seen that one may have two geometries where what is true in one is false in the other, the old opinions such as Plato's, the Schoolmen's, Spinoza's, Leibniz's and Kant's by which mathematics is a science giving completely certain knowledge of existence -- natural or higher-than- natural, and if not of things-in-themselves, then of things as they seem to be -- were not able to be put forward with reason on their side.

The effect of the changed outlook on mathematics and specially geometry was to make theories in mathematics viewed not as in themselves natural knowledge

- 29 - sw ~. .r2:w:ij. iriir .,.4;s' -be -ia.¢|--4-4--4 a hj hlneauU|unineh I . » aiiml iinteflaqugahelpinilnmmmilacieneu. ¢1 . • llldnemnmes I due nnennpm to get lllolutniunnbersandspa1eesandsucl:|like\bings,tonuake inmueatixuug dileovelies about their properties and rehtium, and to put the supported by these dis- ouveries into systems of reasoning. Mathematics is the hee invention of the mind, limited only by possible uses in natural science that may be desired and by rules of logic that one is willing or is forced, as a person of good sense, to be guided by.

The third turning-point in the history of geometry is best talked about in connection with what comes under the fourth and last of the headings given in 2.1, so we will now go on to that.

Sections 5.1 to 6.2 may be okay. 6.3 and 6.4 are bad. 6.5 and 6.6 are good.

- 30 - about it at this stage, though more will be said later. What we will have to say something about here is another side or level of a system of deductions, the idea of which has been very importance for Mathematical Logic.

Some attention in relation to geometry had been given before the 1890's to the question : Is it possible to get the end of one chain of deduction in a system a statement S, which becomes by this fact a theorem -- a true statement -- of the system and to get at the end of another such chain the theorem not-s which says that s is after all, false ? A system of deductions is said to have the property of

He had the same opinion about another question that might be put about a system of deductions. Giving the name

Some thought had been given to this question and the question of consistency from quite early times in connection with attempts to get Euclid's axiom of parallels as a step in a deduction not resting on that axiom, these attempts being made to let one see that that axiom is dependent on the other axioms : which it is not. However, only in the 1890's were these questions regularly put, and put about all systems of deductions in mathematics, and were ways designed for answering them by arguments. Peano and his school were chiefly. responsible for this, though from late in the 1890's the work, in the same direction, that was done by Hilbert in Germany had a greater effect on later developments.

as one might say, the look will be given from the footway at the side of the river and not from the point of view of one having a swim underwater, because clearly we are not able to go deeply into Mathematical Logic here.

What was the start of Mathematical Logic ? The shortest and simplest answer is George Boole's

The earlier teachings in logic of which Boole had a knowledge and which had an effect on him were, on the one hand, those of Sir William Hamilton (1788-1856) and De Morgan (1806-1871) on the theory that was based on the changing of the four, A, E, I, and O forms into a greater number of forms in which the amount of the predicate is given ; for example, Hamilton has two A forms, one being ' all S is all P ' and the other being ' all S is some P '. For later developments two points were important in the theory of Hamilton and De Morgan (the two of whom had a bitter fight, Hamilton protesting that his ideas had been taken from him without his being given credit for them, and De Morgan answering that his account was not at all dependent on anything that Hamilton had ever said). One point - 34 - was the weight placed by it on the amounts of predicates in statements the old logic noting only the amounts of subjects. The value of this was that every statement of the subject-predicate form was able to be turned into an equation or into a statement that an equation was false. By placing this weight on the idea of an equation, logic was moved near to algebra. Secondly, what had been named by Aristotle the two ' ends ', S and P of a statement had generally been viewed as signs. of qualities. In the new theory of Hamilton and De Morgan S and P were changed into being signs of the things themselves that have the qualities. An example of an A statement in the old logic is ' all leaf is green ', this having the sense that the quality of green is a 'part' of the quality of leaf. From the point of view of the new theory this statement had the sense of saying that whatever thing is a leaf has the property of being green: 'all leaves are green things'; or, putting it another way, 'the group of all the things that are leaves is a "part" of all the things that are green'. Putting the change shortly, it was a change from 'all S is P' to 'all S's are P's'. Though the reading of 'all S is P' as 'all S's are P's' had sometimes taken place before Hamilton and De Morgan (for example in Whately's Logak), the reasons for so doing, when there were any conscious reasons, were not reasons of logic. And the greater minds, such as Leibniz's, had kept to the S and P form, not making use of the S's and P's form. It was because they kept to the S and P form that they were stopped from building a better logic. The better logic they had hopes of building was to be a logic of mathematics, even if for more than of mathematics, and it was to be one based on mathematics. However, reasoning in mathematics is generally not about qualities as such but about the things that have the qualities. - 35 - For example, geometry is not a theory of the qualities ' being a point ', ' being a line ' and ' being a plane ', it is a theory of the things that are points, lines and planes .

The name used in logic and mathematics for a group of all the things that have a certain simple or complex property is

But, and this was a great step forward, Boole gave in addition an account of the logic of statement connections (2.7) as an algebra ; unlike Leibniz, Hamilton and De Morgan, he saw that this part of logic was important, aland where others had been completely at a loss he was able to give a good theory of it. It is interesting that his theory was the very same in form as that of the algebra of classes. Boole was the first man to give a united theory of logic.

We have said that Boole 's algebra of statement connections is the very same in form as his algebra of classes. This is true of the body of the book. But after this had been printed an addition of some pages to make clear a number of points in it was put in, and on the very last page Boole made a note of the fact that there is one way in which the two parts of logic are different. The algebra of statement connections has all the laws of algebra of classes and it has one more law in addition.

Boole's algebra of logic is one theory; there are (at least) two systems of readings of it, one in connection with material about classes, the other in connection 4 with material about statements. For example, in the l * v 4 the 2 * 11; i . " -. L *. ... . wisest . 'I . i . .1ii.1{ i M . 4 . . .......Q£_IW 1 .. .. 55 i sa Ur .La f 1 . - i.tun d aliout,th¢8¢ .:. 8 . * $ : ... - ... \ . . . T |"* . | . 1 . .l ..4 ¢ . . . Of . .- froln MI; . . . . being the class having nothing which is an element of - 38 - as ag clenmllla5.-l-y il 'i . : . .. one e]|emenl8 ai°e . n. ; . . * " - | i$ 4W£ A Q ;I 4 l e. e su":. " saw-is .A s. .... .. .. . gg §' \ . ¢4 . .. . i..4 . * .. . m . . u N 4 ts . . 1 . . . .. ? " I 41 1 "l""*"*° \. g * i s y'8 . I * . . . "?* utnat c ¥1 * 8 " *u " . ; l*§ 9 . . . . . . . ; . . E . .I . wgoae e M . 4 93 4 :4 =h=4m1> PJ .. v J.£ ~ 4. I . g.. I 8? . * .;.}- a;||, . e t ul o *.. J, and .if 5 .. .. 34,4 1HMi!l» | . . 4 ... ss...ni*.y..".in.. .. .3 under 4, . . " . . . . . . nl .. . d' ... sign o is. The-Eil&4B4|Ehl=&|Bl|I%g& Q 4 . ."°. .. . 1 . ,wmnh . iS . H44iHm=t@i¢f4!lFIW a . 1 4 onJr-48.91119., butnoQ~th»§wo of them . . . a I *»$a?§lMl'*¥&ui#x "' . 1 *

For Boole a statement 'x' is separate form the statement. ' "x" is true ' in which 'x' is put forward as being true ; the second statement is separate from the first if only because the first might be put forward as being false and not as being true. In Boole's algebras of statement connections the statement ' "x" is true ' becomes the equation x = 1, while the statement ' "x" is false ' becomes the equation x = 0. Because every x is true or false, a law of this algebra is: x = 0 or x = 1. This law is specific to the algebra of statement connections ( see end of 6.4).

In

On the other point, about time, Boole says : let us make use of the letters

One of the chief points made by Boole in

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