## 1. Career




Born in 1809, Peirce became a major figure in mathematics and the
physical sciences during a period when the U.S. was still a minor
country in these areas (Hogan 1991). A student at Harvard College, he
was appointed tutor there in 1829. Two years later he became Professor
of Mathematics in the University, a post which was changed in 1842 to
cover astronomy also; he held it until his death in 1880. He played a
prominent role in the development of the science curriculum of the
university, and also acted as College librarian for a time. However, he
was not a successful teacher, being impatient with students lacking
strong gifts; but he wrote some introductory textbooks in mathematics,
and also a more advanced one in mechanics (Peirce 1855). Among his
other appointments, the most important one was Director of the U.S.
Coast Survey from 1867 to 1874. Peirce also exercised influence through
his children. By far the most prominent was
 Charles Sanders Peirce
 (1839-1914), who became a remarkable though maverick polymath,
as mathematician, chemist, logician, historian, and many other
activities. In addition, James Mills (1834-1906) became in turn
professor of mathematics at Harvard, Benjamin Mills (1844-1870)
a mining engineer, and Herbert Henry Davis (1849-1916) a
diplomat. Harvard professor Benjamin Osgood Peirce (1854-1914),
mathematician and physicist was a cousin. Benjamin Peirce did not
think of himself as a philosopher in any academic sense, yet his work
manifests interests of this kind, in two different ways. The first was
related to his teaching. 


## 2. Mathematics, mechanics and God




To a degree unusually explicit in a mathematician of that time Peirce
affirmed his Christianity, seeing mathematics as study of God's work by
God's creatures. He rarely committed such sentiments to print; but a
short passage occurs in the textbook on mechanics previously mentioned,
when considering the idea that the occurrence of perpetual motion in
nature 



> would have proved destructive to human belief, in the
> spiritual origin of force and the necessity of a First Cause superior
> to matter, and would have subjected the grand plans of Divine
> benevolence to the will and caprice of man (Peirce 1855,
> 31).




Peirce was more direct in a course of Lowell Lectures on
'Ideality in the physical sciences' delivered at Harvard in
1879, which James Peirce edited for posthumous publication (Peirce
1881b). 'Ideality' connoted 'ideal-ism' as
evident in certain knowledge, 'pre-eminently the foundation of
the mathematics'. His detailed account concentrated almost
entirely upon cosmology and cosmogony with some geology (Petersen
1955). He did not argue for his stance beyond some claims for existence
by design. 


## 3. Algebras and their philosophy




Peirce was primarily an algebraist in his mathematical style; for
example, he was enthusiastic for the cause of quaternions in mechanics
after their introduction by W. R. Hamilton in the mid 1840s, and of the
various traditions in mechanics he showed some favour for the
'analytical' approach, where this adjective refers to the
links to algebra. His best remembered publication was a treatment of
'linear associative algebras', that is, all algebras in
which the associative law
*x*(*yz*)=(*xy*)*z* was upheld.
'Linear' did not carry the connotation of matrix theory,
which was still being born in others' hands, but referred to the form
of linear combination, such as: 



> *q = a + bi + cj + dk*




in the case of a quaternion *q*. Peirce wrote an extensive
survey (Peirce 1870), determining the numbers of all algebras with from
two to six elements obeying also various other laws (Walsh 2000, ch.
2). To two of those he gave names which have become durable:
'idempotent', the law *x**m* =
*x* (for *m*[?]2) which George Boole had introduced in
this form in his algebra of logic in 1847; and 'nilpotent',
when *x**m* = 0, for some *m*. The
history of the publication of this work is very unusual
(Grattan-Guinness 1997). Peirce had presented some of his results from
1867 onwards to the National Academy of Sciences, of which he had been
appointed a founder member four years earlier; but they could not
afford to print it. Thus, in an initiative taken by Coast Survey staff,
a lady without mathematical training but possessing a fine hand was
found who could both read his ghastly script and write out the entire
text 12 pages at a time on lithograph stones. 100 copies were printed
(Peirce 1870), and distributed world-wide to major mathematicians and
professional colleagues. Eleven years later Charles, then at Johns
Hopkins University, had the lithograph reprinted posthumously, with
some additional notes of his own, as a long paper in American journal
of mathematics, which J.J. Sylvester had recently launched (Peirce
1881a); it also came out in book form in the next year. This study
helped mathematicians to recognise an aspect of the wide variety of
algebras which could be examined; it also played a role in the
development of model theory in the U.S. in the early 1900s. Enough work
on it had been done by then for a book-length study to be written (Shaw
1907). 


## 4. The philosophy of necessity




Peirce seems to have upheld his theological stance for all mathematics,
and a little sign is evident in the dedication at its head: 



> To my friends This work has been the pleasantest
> mathematical effort of my life. In no other have I seemed to myself to
> have received so full a reward for my mental labor in the novelty and
> breadth of the results. I presume that to the uninitiated the formulae
> will appear cold and cheerless. But let it be remembered that, like
> other mathematical formulae, they find their origin in the divine
> source of all geometry. Whether I shall have the satisfaction of taking
> part in their exposition, or whether that will remain for some more
> profound expositer, will be seen in the future (Peirce 1870,
> 1).




Peirce began with a philosophical statement of a different kind about
mathematics which has become his best remembered single statement
"Mathematics is the science that draws necessary conclusions" (Peirce
1870, p. 1). What does 'necessary' denote? Perhaps he was
following a tradition in algebra, upheld especially by Britons such as
George Peacock and Augustus De Morgan (a recipient of the lithograph),
of distinguishing the 'form' of an algebra from its
'matter' (that is, an interpretation or application to a
given mathematical and/or physical situation) and claiming that its
form alone would deliver the consequences from the premises. In his
first draft of his text he wrote the rather more comprehensible
"Mathematics is the science that draws inferences", and in the second
draft "Mathematics is the science that draws consequences", though the
last word was altered to yield the enigmatic form involving
'necessary' used in the book. The change is not just
verbal; he must have realised that the earlier forms were not
sufficient (they are satisfied by other sciences, for example), and so
added the crucial adjective. Certainly no whiff of modal logic was in
his air. His statement appears in the mathematical literature fairly
often, but usually without explanation. One feature is clear, but often
is not stressed. In all versions Peirce always used the active verb
'draws': mathematics was concerned with the act of drawing
conclusions, not with the theory of so acting, which belonged in
disciplines such as logic. He continued: 



> Mathematics, as here defined, belongs to every enquiry;
> moral as well as physical. Even the rules of logic, by which it is
> rigidly bound could not be deduced without its aid (Peirce 1870,
> 3).




In a lecture of the late 1870s he described his definition as 



> wider than the ordinary definitions. It is subjective; they
> are objective. This will include knowledge in all lines of research.
> Under this definition mathematics applies to every mode of enquiry
> (Peirce 1880, 377).




Thus Peirce maintained the position asserted by Boole that mathematics
could be used to analyse logic, not the vice versa relationship between
the two disciplines that Gottlob Frege was about to put forward for
arithmetic, and which Bertrand Russell was optimistically to claim for
*all* mathematics during the 1900s. Curiously, the third draft
of the lithograph contains this contrary stance in "Mathematics, as
here defined, belongs to every enquiry; it is even a portion of
deductive logic, to the laws of which it is rigidly subject"; but by
completion he had changed his mind. Peirce's son Charles claimed to
have influenced his father in forming his definitive position, and
fiercely upheld it himself; thereby he helped to forge a wide division
between the algebraic logic which he was developing from the early
1870s with his father, Boole and de Morgan as chief formative
influences, and the logicism (as it became called later) of Frege and
Russell and also the 'mathematical logic' of Giuseppe Peano
and his school in Turin (Grattan-Guinness 1988).