% LaTeX source for Macfarlane's lecture on Boole





  \Large{GEORGE BOOLE}\footnote{This Lecture was delivered April 
  19, 1901.---\textsc{Editors}}\\

\textsc{George Boole} was born at Lincoln, England, on the 2d of
November 1815. His father, a tradesman of very limited means, was
attached to the pursuit of science, particularly of mathematics, and was
skilled in the construction of optical instruments. Boole received his
elementary education at the National School of the city, and afterwards
at a commercial school; but it was his father who instructed him in the
elements of mathematics, and also gave him a taste for the construction
and adaptation of optical instruments. However, his early ambition did
not urge him to the further prosecution of mathematical studies, but
rather to becoming proficient in the ancient classical languages. In
this direction he could receive no help from his father, but to a
friendly bookseller of the neighborhood he was indebted for instruction
in the rudiments of the Latin Grammar. To the study of Latin he soon
added that of Greek without any external assistance; and for some years
he perused every Greek or Latin author that came within his reach. At
the early age of twelve his proficiency in Latin made him the occasion
of a literary controversy in his native city. He produced a metrical
translation of an ode of Horace, which his father in the pride of his
heart inserted in a local journal, stating the age of the translator. A
neighboring school-master wrote a letter to the journal in which he
denied, from internal evidence, that the version could have been the
work of one so young. In his early thirst for knowledge of languages and
ambition to excel in verse he was like Hamilton, but poor Boole was much
more heavily oppressed by the \textit{res angusta domi}---the hard
conditions of his home. Accident discovered to him certain defects in
his methods of classical study, inseparable from the want of proper
early training, and it cost him two years of incessant labor to correct

Between the ages of sixteen and twenty he taught school as an assistant
teacher, first at Doncaster in Yorkshire, afterwards at Waddington near
Lincoln; and the leisure of these  cars he devoted mainly to the study
of the principal modern languages, and of patristic literature with the
view of studying to take orders in the Church. This design, however, was
not carried out owing to the financial circumstances of his parents and
some other difficulties. In his twentieth year he decided on opening a
school on his own account in his native city; thenceforth he devoted all
the leisure he could command to the study of the higher mathematics,
and solely with the aid of such books as he could procure. Without other
assistance or guide he worked his way onward, and it was his own opinion
that he had lost five years of educational progress by his imperfect
methods of study, and the want of a helping hand to get him over.
difficulties. No doubt it cost him much time; but when he had finished
studying he was already not only learned but an experienced
investigator.  We have seen that at this time (1835) the great masters
of mathematical analysis wrote in the French language; and Boole was
naturally led to the study of the \textit{M\'ecanique c\'eleste} of
Laplace, and the \textit{M\'ecanique analytique} of Lagrange. While
studying the latter work he made notes from which there eventually
emerged his first mathematical memoir, entitled, ``On certain theorems
in the calculus of variations.'' By the same works his attention was
attracted to the transformation of homogeneous functions by linear
substitutions, and in the course of his subsequent investigations he was
led to results which are now regarded as the foundation of the modern
Higher Algebra. In the publication of his results he received friendly
assistance from  D.~F.~ Gregory, a younger member of the Cambridge
school, and editor of the newly founded Cambridge Mathematical Journal.
Gregory and other friends suggested that Boole should take the regular
mathematical course at Cambridge, but this he was unable to do; he
continued to teach school for his own support and that of his aged
parents, and to cultivate mathematical analysis in the leisure left by a
laborious occupation.        

Duncan~F.~Gregory was one of a Scottish family already distinguished in
the annals of science. His grandfather was James Gregory, the inventor
of the refracting telescope and discoverer of a convergent series for
$\pi$. A cousin of his father was David Gregory, a special friend and
fellow worker of Sir Isaac Newton. D. F. Gregory graduated at Cambridge,
and after graduation he immediately turned his attention to the logical
foundations of analysis. He had before him Peacock's theory of algebra,
and he knew that in the analysis as developed by the French school there
were many remarkable phenomena awaiting explanation; particularly theorems
which involved what was called the separation of symbols. He embodied his
results in a paper ``On the real Nature of symbolical Algebra'' which
was printed in the \textit{Transactions} of the Royal Society of

Boole became a master of the method of separation of symbols, and by
attempting to apply it to the solution of differential equations with
variable coefficients was led to devise a general method in analysis.
The account of it was printed in the \textit{Transactions} of the Royal
Society of London, and brought its author a Royal medal. Boole's study
of the separation of symbols naturally led him to a study of the
foundations of analysis, and he had before him the writings of Peacock,
Gregory and De Morgan. He was led to entertain very wide views of the
domain of mathematical analysis; in fact that it was coextensive with
exact analysis, and so embraced formal logic. In 1848, as we have seen
the controversy arose between Hamilton and De Morgan about the
quantification of terms; the general interest which that controversy
awoke in the relation of mathematics to logic induced Boole to prepare
for publication his views on the subject, which he did that same year in
a small volume entitled \textit{Mathematical Analysis of Logic}. 

About this time what are denominated the Queen's Colleges of Ireland
were instituted at Belfast, Cork and Galway; and in 1849 Boole was
appointed to the chair of mathematics in  the Queen's College at Cork.
In this more suitable environment he set himself to the preparation of a
more elaborate work on the mathematical analysis of logic. For this purpose
he read extensively books on psychology and logic, and as a result
published in 1854 the work on which his fame chiefly rests---``An
Investigation of the Laws of Thought, on which are founded the
mathematical theories of logic and probabilities.'' Subsequently he
prepared textbooks on \textit{Differential Equations} and \textit{Finite
Differences}\,; the former of which remained the best English textbook on
its subject until the publication of Forsyth's \textit{Differential

Prefixed to the Laws of Thought is a dedication to Dr.\ Pyall,
Vice-President and Professor of Greek in the same College. In the
following year, perhaps as a result of the dedication, he married Miss
Everest, the niece of that colleague. Honors came: Dublin University
made him an LL.D., Oxford a D.C.L.; and the Royal Society of London
elected him a Fellow. But Boole's career was cut short in the midst of
his usefulness and scientific labors. One day in 1864 he walked from his
residence to the College, a distance of two miles, in a drenching rain,
and lectured in wet clothes. The result was a feverish cold which soon
fell upon his lungs and terminated his career on December 8, 1864, in
the 50th year of his age.  

De Morgan was the man best qualified to judge of the value of Boole's
work in the field of logic; and he gave it generous praise and help. In
writing to the Dublin Hamilton he said,  ``I shall be glad to see his
work (\textit{Laws of Thought}) out, for he has, I think, got hold of
the true connection of algebra and logic.'' At another time he wrote to
the same as follows:  ``All metaphysicians except you and I and Boole
consider mathematics as four books of Euclid and algebra up to quadratic
equations.'' We might infer that these three contemporary mathematicians
who were likewise philosophers would form a triangle of friends. But it
was not so; Hamilton was a friend of De Morgan, and De Morgan a friend
of Boole; but the relation of friend; although convertible, is not
necessarily transitive. Hamilton met De Morgan only once in his life,
Boole on the other hand with comparative frequency; yet he had a
voluminous correspondence with the former  extending Over 20 years, but
almost no correspondence with the latter. De Morgan's investigations of
double algebra and triple algebra prepared him to appreciate the
quaternions, whereas Boole was  too much given over to the symbolic
theory  to appreciate geometric algebra. 

Hamilton's biography has appeared in three volumes, prepared by his
friend Rev.\ Charles Graves; De Morgan's biography has appeared in one
volume, prepared by his widow; of Boole no biography has appeared. A
biographical notice of Boole was written for the \textit{Proceedings} of
the Royal Society of London by his friend the Rev. Robert Harley, and it
is to it that I am indebted for most of my biographical data. Last
summer  when in England I learned that the reason why no adequate
biography of Boole had appeared was the unfortunate temper and lack of
sound judgment of his widow. Since her husband's death Mrs.\ Boole has
published a paradoxical book of the false kind worthy of a notice in De
Morgan's \textit{Budget}. 

The work done by Boole in applying mathematical analysis to logic
necessarily led him to consider the general question of how reasoning is
accomplished by means of symbols. The view which he adopted on this
point is stated at page 68 of the \textit{Laws of Thought}.  ``The
conditions of valid reasoning by the aid of symbols, are:
\textit{First}, that a fixed interpretation be assigned to the symbols
employed in the expression of the data; and that the laws of the
combination of these symbols be correctly determined from that
interpretation; \textit{Second}, that the formal processes of solution
or demonstration be conducted throughout in obedience to all the laws
determined as above, without regard to the question of the
interpretability of the particular results obtained; \textit{Third},
that the final result be interpretable in form, and that it be actually
interpreted in accordance with that system of interpretation which has
been employed in the expression of the data.'' As regards these
conditions it may be observed that they are very different from the
formalist view of Peacock and De Morgan, and that they incline towards a
realistic view of analysis, as held by Hamilton. True he speaks of
interpretation instead of meaning, but it is a fixed interpretation; and
the rules for the processes of solution are not to be chosen
arbitrarily, but are to be found out from the particular system of
interpretation of the symbols.  

It is Boole's second condition which chiefly calls for study and
examination; respecting it he observes as follows: ``The principle in
question may be considered as resting upon a general law of the mind
the knowledge of which is not given to us \textit{a priori}, that is,
antecedently to experience, but is derived, like the knowledge of the
other laws of the mind, from the clear manifestation of the general
principle in the particular instance. A single example of reasoning, in
which symbols are employed in obedience to laws founded upon their
interpretation, but without any sustained reference to that
interpretation, the chain of demonstration conducting us through
intermediate steps which are not interpretable to a final result which
is interpretable, seems not only to establish the validity of the
particular application, but to make known to us the general law
manifested therein. No accumulation of instances can properly add
weight to such evidence.  It may furnish us with clearer conceptions of
that common element of truth upon which the application of the principle
depends, and so prepare the way for its reception. It may, where the
immediate force of the evidence is not felt, serve as a verification,
\textit{a posteriori}, of the practical validity of the principle in
question. But this does not affect the position affirmed, viz., that
the general principle must be seen in the particular instance---seen to
be general in application as well as true in the special ex-ample. The
employment of the uninterpretable symbol $\sqrt{-1}$ the intermediate
processes of trigonometry furnishes an illustration of what has been
said. I apprehend that there is no mode of explaining that application
which does not covertly assume the very principle in question. But that
principle, though not as I conceive, warranted by formal reasoning
based upon other grounds, seems to deserve a place among those axiomatic
truths which constitute in some sense the foundation of general
knowledge, and which may properly be regarded as expressions of the
mind's own laws and constitution.''  

We are all familiar with the fact that algebraic reasoning may be
conducted through intermediate equations without requiring a sustained
reference to the meaning of these equations; but it is paradoxical to
say that these equations can, in any case, have no meaning or
interpretation. It may not be necessary to consider their meaning, it
may even be difficult to find their meaning, but that they have a
meaning is a dictate of common sense. It is entirely paradoxical to say
that, as a general process, we can start from equations having a
meaning, and arrive at equations having a meaning by passing through
equations which have no meaning. The particular instance in which
Boole sees the truth of the paradoxical principle is the successful
employment of the uninterpretable symbol $\sqrt{-1}$ in the intermediate
processes of trigonometry. So  soon then as this symbol is interpreted,
or rather, so soon as its meaning is demonstrated, the evidence for the
principle fails, and Boole's transcendental logic falls.  

In the algebra of quantity we start from elementary symbols denoting
numbers, but are soon led to compound forms which do not reduce to
numbers; so in the algebra of logic we start from elementary symbols
denoting classes, but are soon introduced to compound expressions which
cannot be reduced to simple classes. Most mathematical logicians say,
Stop, we do not know what this combination means. Boole says, It may be
meaningless, go ahead all the same. The design of the \textit{Laws of
Thought} is stated by the author to be to investigate the fundamental
laws of those operations of the mind by which reasoning is performed; to
give expression to them in the symbolical language of a Calculus, and
upon this foundation to establish the Science of Logic and construct its
method; to make that method itself the basis of a general method for the
application of the mathematical doctrine of Probabilities; and, finally
to collect from the various elements of truth brought to view in the
course of these inquiries some probable intimations concerning the
nature and constitution of the human mind.  

Boole's inventory of the symbols required in the algebra of logic is as
follows: \textit{first}, Literal symbols, as $x$, $y$, etc.,
representing things as subjects of our conceptions; \textit{second},
Signs of operation, as $+$, $-$, $\times$, standing for those operations
of the mind by which the conceptions of things are combined or resolved
so as to form new conceptions involving the same elements;
\textit{third}, The sign of identity $=$; not equality merely, but
identity which involves equality. The symbols $x$, $y$, etc., are used
to denote classes; and it is one of Boole's maxims that substantives and
adjectives alike denote classes.  ``They may be regarded,'' he says,
``as differing only in this respect, that the former expresses the
substantive existence of the individual thing or things to which it
refers, the latter implies that existence. If we attach to the adjective
the umversally understood subject, ``being'' or ``thing,'' it becomes
virtually a substantive, and may for all the essential purposes of
reasoning be replaced by the substantive.''  Let us then agree to
represent the class of individuals to which a particular name is
applicable by a single letter as $x$. If the name is \textit{men} for
instance, let $x$ represent \textit{all men}, or the class
\textit{men}.' Again, if an adjective, as \textit{good}, is employed as
a term of description, let us represent by a letter, as $y$, all things
to which the description \textit{good} is applicable, that is,
\textit{all good things} or the class \textit{good things}. Then the
combination $yx$ will represent good men.  

Boole's symbolic logic was brought to my notice by Professor Tait,
when I was a student in the physical laboratory of Edinburgh University.
I studied the \textit{Laws of Thought} and I found that those who had
written on it regarded the method as highly mysterious; the results
wonderful, but the processes obscure. I reduced everything to diagram
and model, and I ventured to publish my views on the subject in a small
volume called \textit{Principles of the Algebra of Logic}; one of the
chief points I made is the philological and analytical difference
between the substantive and the adjective. What I said was that the 
word \textit{man} denotes a class, but the word \textit{white} does not;
in the former a definite unit-object is specified, in the latter no
unit-object is specified. We can exhibit a type of a \textit{man} we
cannot exhibit a type of a \textit{white}.  

The identification of the substantive and adjective on the one hand and
their discrimination on the other hand lead to different conceptions of
what De Morgan called the universe. Boole's conception of the Universe
is as follows (\textit{Laws of  Thought}, p.\ 42): ``In every discourse,
whether of the mind conversing with its own thoughts, or of the
individual in his intercourse with others, there is an assumed or
expressed limit within which the subjects of its operation are confined.
The most unfettered discourse is that in which the words we use are
understood in the widest possible application, and for them the limits
of discourse are coextensive with those of the universe itself. But more
usually we confine ourselves to a less spacious field. Sometimes in
discoursing of men we imply (without expressing the limitation) that it
is of men only under certain circumstances and conditions that we speak,
as of civilized men, or of men in the visor of life, or of men under
some other condition or relation. Now, whatever may be the extent of the
field within which all the objects of our discourse are found, that
field may properly be termed the universe of discourse.''

Another view leads to the conception of the Universe as a collection of
homogeneous units, which may be finite or infinite in number; and in a
particular problem the mind considers the relation of identity between
different groups of this collection. This universe corresponds to
\textit{the series of events} in the theory of Probability; and the
characters correspond to the different ways in which the event may
happen. The difference is that the Algebra of Logic considers necessary
data and relations. while the theory of Probability considers probable
data and relations. I will explain the elements of Boole's method on
this  theory. 

The square is a collection of points: it may serve to represent any
collection of homogeneous units, whether finite or infinite  in number, 
that is, the universe of the problem. Let $x$ denote \textit{inside the
left-hand circle}, and $y$ \textit{inside the right-hand circle}. $Uxy$
will denote the points inside both circles $y$ (Fig.\ 1). In
arithmetical value $x$ may range from 1 to 0; so also $y$; while $xy$
cannot be greater than $x$ or $y$, or less than 0 or $x+y-1$. This last is   
the  principle of the syllogism. From the co-ordinate  nature of the
operations $x$ and $y$, it is evident that $Uxy = Uyx$; but this is a
different thing from commuting, as Boole does, the relation of $U$ and $x$,
which is not that of co-ordination, but of subordination of $x$ to $U$, and
which is Properly denoted by writing $U$ first.  

Suppose $y$ to be the same character as $x$; we will then always have
$Uxx = Ux$; that is, an elementary selective symbol $x$ is always such
that $x^2=x$. These are but the symbols of ordinary algebra which
satisfy this relation, namely 1 and 0; these are also the extreme
selective symbols all and none. The law in question was considered
Boole's paradox. it plays a very great part in the development of his

Let $Uxy = Uz$, where = means identical with, not equal to; we may write
$xy = z$, leaving the $U$ to be understood. It does not mean that the
combination of characters xy is identical with the character $z$; but
that those points which have the characters $x$ and $y$ are identical
with the  points which have the character $z$ (Fig.\ 2). From $xy = z$,
we derive $x = \frac{1}{y}z$; what is the meaning of this expression? We
shall return to the question, after we have considered + and $-$.

Let us now consider the expression $U(x+y)$. If the $x$ points and the
$y$ points are outside of one another, it means the sum of the $x$
points and the points (Fig.\ 3) So far all are agreed.  But suppose that
the $x$ points and the $y$ points are partially identical  (Fig. 4);
then there arises difference of opinion.  Boole held that the common
points must be taken twice over, or in other words that the symbols $x$
and $y$ must be treated all the same as if they were independent of one
another; otherwise, he held, no general analysis is possible.  $U(x+y)$
will not in general denote a single class of points; it will involve in
general a duplication.  

Similarly, Boole held that the expression $U(x-y)$ does not involve the
condition of the $Uy$ being wholly included in the $Ux$ (Fig.\ 5). If
that condition is satisfied $U(x-y)$ denotes a simple class; namely, the
$Ux$'s \textit{without} the $Uy$'s. But when there is partial
coincidence (as in Fig.\ 4), the common points will be cancelled, and
the result will be the $Ux$'s which are not $y$ taken positively and the
$Uy$'s which are not $x$ taken negatively. In Boole's view $U(x-y)$ was
in general an intermediate uninterpretable form, which might be used in
reasoning the same way as analysts used $\sqrt{-1}$.   

Most of the mathematical logicians who have come after Boole are men who
would have stuck at the impossible subtraction in ordinary algebra. They
say virtually, ``How can you throw into a heap the same things twice
over; and how can you take from a heap things that are not there.''
Their great principle is the impossibility of taking the pants from a
Highlander. Their only conception of the analytical processes of
addition and subtraction is throwing into a heap and taking out of a
heap. It does not occur to them that the processes of algebra are
\textit{ideal}, and not subject to gross material restrictions.  

If $x+y$ denotes a quality without duplication, it will satisfy the
  (x+y)^2 &= x + y, \\
  x^2 + 2xy + y^2 &= x+y
  x^2=x\qquad y^2 &= y \\
  \therefore 2xy &= 0.
Similarly, if $x-y$ denote a simple quality, then 
  (x-y)^2 &= x-y, \\
  x^2+y^2  = 2xy  &= x-y, \\
  x^2 = x,\qquad y^2 &= y, \\
  y-2xy &= -y, \\
  \therefore y &=xy.
In other words, the $Uy$ must be included in the $Ux$ (Fig.\ 5). Here we
have assumed that the law of signs is the same as in ordinary algebra,
and the result comes out correct.  

Suppose $Uz = Uxy$; then $Ux = U\frac{1}{y}z$.  How are the $Ux$'s
related to the $Uy$'s and the $Uz$'s?  From the diagram (in Fig.\ 2) we
see that the $Ux$'s are identical with all the $Uyz$'s together with an
indefinite portion of the $U$'s, which are neither $y$ nor $z$. Boole
discovered a general method for finding the meaning of any function of
elementary logical symbols, which applied to the above case, is as
follows.  When $y$ is an elementary symbol,  
  1 &= y+(1-y). \\
  1 &= z+(1-z). \\
  \therefore 1 &= yz+y(1-z)+(1-y)z+(1-y)(1-z), 
which means that the $U$'s either have both qualities $y$ and $z$, or 
$y$ but not $z$, or $z$ but not $y$, or neither $y$ and $z$. Let  
\[ \frac{1}{y}z = Ayz + By(1 -z) + C(1-y)z + D(1-y)(1-z), \] 
it is required to determine the coefficients $A$, $B$, $C$, $D$.
Suppose $y = 1,$ $z = 1$; then $1 = A$. Suppose $y = 1$, $z = 0$, 
then $0 = B$.  Suppose $y = 0$, $z = 1$; then $\frac{1}{0} = C$, 
and $C$ is infinite; therefore $(1-y)z = 0$; which we see to be true 
from the diagram. Suppose $y = 0$, $z = 0$; then $\frac{0}{0} = D$, 
or $D$ is indeterminate. Hence 
  $\frac{1}{y}z = yz + $an indefinite portion of $(1 - y)(1 - z)$. 
  *\qquad*\qquad*\qquad*\qquad*\qquad* \end{center} Boole attached great
importance to the index law $x^2=x$. He held that it expressed a law of
thought, and formed the characteristic distinction of the operations of
the mind in its ordinary discourse and reasoning, as compared with its
operations when occupied with the general algebra of quantity. It makes
possible, he said, the solution of a quintic or equation of higher
degree, when the symbols are logical. He deduces from it the axiom of
metaphysicians which is termed the principle of contradiction, and which
affirms that it is impossible for any being to possess a quality, and at
the same time not to possess it. Let $x$ denote an elementary quality
applicable to the universe $U$; then $1-x$ denotes the absence of that 
quality. But if $x^2=x$, then $0=x-x^2$, $0=x(1-x)$, that is, from 
$Ux^2=Ux$ we deduce $Ux(1-x)=0$. 

He considers $x(1-x)=0$ as an expression of the principle of
contradiction. He proceeds to remark: ``The above interpretation has
been introduced not on account of its immediate value in the present
system, but as an illustration of a significant fact in the philosophy
of the intellectual powers, viz., that what has been commonly regarded
as the fundamental axiom of metaphysics is but the consequence of a
law of thought, mathematical in its form. I desire to direct attention
also to the circumstance that the equation in which that fundamental law
of thought is expressed is an equation of the second degree. Without
speculating at all in this chapter upon the question whether that
circumstance is necessary in its own nature, we may venture to assert
that if it had not existed, the whole procedure of the understanding
would have been different from what it is.''  

We have seen that De Morgan investigated long and published much on
mathematical logic. His logical writings are characterized by a display
of many symbols, new alike to logic and to mathematics; in the words of
Sir W.~Hamilton of Edinburgh, they are ``horrent with mysterious
spicul\ae.'' It was the great merit of Boole's work that he used the
immense power of the ordinary algebraic notation as an exact language.
and proved its power for making ordinary language more exact. De Morgan
could well appreciate the magnitude of the feat, and he gave generous
testimony to it as follows: 


``Boole's system of logic is but one of many proofs of genius  and
patience combined. I might legitimately have entered it among, my
\textit{paradoxes}, or things counter to general opinion: but it is a
paradox which, like that of Copernicus, excited admiration from its
:first appearance. That the symbolic processes of algebra, invented as
tools of numerical calculation, should be competent to express every
act of thought, and to furnish the grammar and dictionary of an
all-containing system of logic, would not have been believed until it
was proved. When Hobbes, in the time of the Commonwealth, published his
``Computation or Logique'' he had a remote glimpse of some of the points
which are placed in the light of day by Mr. Boole. The unity of the
forms of thought in all the applications of reason, however remotely
separated, will one day be matter of notoriety and common wonder: and
Boole's name will be remembered in connection with one of the most
important steps towards the attainment of this knowledge.''
  From A~Macfarlane, \textit{Lectures on Ten British Mathematicians of the
  Nineteenth Century}, New York: Wiley and London: Chapman and Hall 1916,