MATHEMATICAL LOGIC: MATHEMATICS OF LOGIC OR LOGIC OF MATHEMATICS
Z. Šikić
This brief historical survey, written from the logical point of view, is a rational
reconstruction of the genesis of some interrelations between formal logic and mathematics.
We start with the period which preceded the important logical writings of Boole.
At the end of the first half of 19 th century formal logic was being developed mainly in
Great Britain, under the influence of the newly raising abstract algebra, which have just begun
to appear in the papers of G. Peacock, J. Gregory, A. de Morgan and W. R. Hamilton.
Attempts to apply mathematical analysis (which proved so fruitful and convincing in studying
the laws of Quality) to formal logic (which deals with the laws of Quality) became
characteristic for that period. Setting a suitable symbolic apparatus and founding the laws of
its manipulation, in the same way arithmetic does it, was the ultimate purpose of the use of the
new method. Similar attempts originated from earlier periods and other places (e. g. G. W.
Leibniz on continent) but the first great success was achieved in 1847 by Boole’s
“Mathematical Analysis of Logic”. It was Boole who showed that the fundamental operations
with concepts (operations of thought, he would say) can be represented by arithmetic
operations, addition, subtraction and multiplication; while the fundamental concepts of
Everything and Nothing can be represented by 1 and 0. Namely, these operations are
governed by the arithmetic laws (i. e. by the laws of operations with quantities) together with
an additional one: x2 = x (called the principle of tautology) characteristic and distinctive for
operations with qualities. Thus the logic of concepts (or classes) became special arithmetic
and it is in this way that G. Boole understands it. What is a conclusion? Premises are
equations which, using arithmetic operations, can produce other equations, the latter being
thus conclusions of the arithmetized reasoning. The problem of the logical interpretation of
the obtained equations is solved by reducing the equation to its normal form, applying
Taylor’s formula characteristic for this (x2 = x)-arithmetic: f(x) = f(1)x – f(0)(1 – x). By this
analysis the logic of concepts acquired simplicity, safety and generality of arithmetic. In other
words, it was well, and that meant arithmetically, founded.
Equations which was not possible to interpret in the course of the proof were a
disturbance to the purely logical understanding of the basic laws, and hence there began dearithmetizing of formal logic (i. e. G. Boole himself, W. S. Jevons, S. C. Pierce, J. Venn, E.
Schröder)1). The final result was a Boole-Schroeder algebra which represented, in its various
interpretations, the logic of concepts, the logic of one-place propositional functions (from G.
Boole on), as well as the logic of propositions (from C. S. Pierce and E. Schröder on). It
should be noted that terms propositional function and proposition bring along views and
contents which have not been treated yet in this period (see 2)). Bearing in mind the period we
have been trying to analyse it is better to say that the logic of primary and secondary
propositions or categorical and hypothetical syllogisms in particular, obey the same BooleSchröder’s laws.
De-arithmetizing of formal logic has not basically changed the character of its
founding (which is often thought and said on the basis of the later logistic or formalistic
understanding of the reduction of mathematics) but it has substantially influenced the
character of mathematics. Namely, Boole-Schröder’s algebra has been founded as an abstract
mathematical system, independent of its particular interpretation, by the method of deduction
from a small number of premises, the method which in itself becomes a characteristic of
mathematics regardless of the quantitative or any other character of the matter being
researched. However, the method itself becomes a paradox if the system, which is being
founded, is formal logic itself: mathematical deductions are unavoidably based on the laws
which are yet to be founded in formal logic. Consequently, the result of purifying formal logic
by de-arithmetizing it, was not a pure formal logic, but rather a pure mathematics. In this
sense, G. Boole is the father of pure mathematics and impure mathematical logic2).
This discussion bring us to the new epoch in the development of formal logic and
mathematics, and hereby we end the survey of the epoch which originally considers formal
logic as a special arithmetic, then as a special case of pure mathematics and which
accordingly introduces the term mathematical analysis of logic, reminiscent of the term
mathematical logic, if we read it as mathematics of logic.
The criticism of the infinitesimal calculus and the return of mathematics to the
problem of its foundation in 1850-s, clearly stated the need for the foundation of the
arithmetic of real numbers. It is a period of the critical movement in mathematics, a period
when K. Weirstrass became a supreme mathematical authority (“Notre maître à tous”, said
Hermite), a period where the intuition of space and time is being rejected, by informal logical
analysis, as a basis of arithmetic. In this period R. Dedekind arithmetizes the continuum of
real numbers3) and further sets a problem of natural number genesis, saying: “When I say that
arithmetic (algebra, analysis) is only a part of logic, I already state that I consider the concept
of number completely independent of the notions or intuitions of space and time, and, on the
contrary, I consider it to be a direct product of the pure laws of reasoning” 4). It is the
beginning of the epoch which does not have a good ear for the logic which looks for support
in arithmetic, but on the contrary, it looks for support of arithmetic (and mathematics) in
logic. However, this radical turning point was gradually prepared by adopting and developing
one more aspect of the critical movement in mathematics. In the mind of the leading
mathematicians at the end of 19th century, there is a clear-cut ideal of a mathematical theory
which is derived from the small number of (mathematical) premises in accordance with the
logical principles; the ideal of the theory which rejects intuition even as a means of
demonstration. Some more radical thinkers like G. Peano realized that the deductive science
understood in this way (which demands postulation and definition of its basic entities 5) but
also a radical cleansing of the deductive process from the influence of intuition) can be
formulated only within the frame of a symbolic language which is freed from the intuitive
content of natural languages. Mathematics has already developed such a language to a great
extent. However, G. Peano thinks that it is necessary (according to the new concept of the
deductive science) to do something that has not been done even in mathematics: to formalize
and symbolically describe the very arguments of mathematics. With this purpose in mind he
designed mathematical logic and used it in the formalization of mathematics 6). Due to this
mathematical logic, the deduction of a conclusion from premises was replaced by formal
generation of the respective symbolic expression, from other expressions of that kind, by a
quasi-algebraic process7). Hence, mathematics was freed from intuition, becoming a set of
propositions in the form of “p implies q”, where p represents a conjunction of the postulated
mathematical assertions (axioms) and q their quasi-algebraically (mechanically) derived
consequence8).
The connection between this aspect of the critical movement with the one represented
by Dedekind, who stated that the basic constituents of mathematical propositions (natural
number, integer, rational number, real number) are logically definable 9), draws attention to the
possibility of p and q being purely logical propositions (propositions made up of purely
logical concepts), i.e. that mathematics, freed from intuition, has become a set of purely
logical propositions in the form of “p implies q”; in other words, that mathematics has
become a part of logic. Joining these two aspects of the critical movement in mathematics and
a consistent acceptance of such an attitude is to the merit of B. Russell 10). In this way, a
logicist epoch in the apprehension of the relations of formal logic and mathematics has been
initiated, an epoch which considers arithmetic, and consequently all pure mathematics,
as a part of logic and which accordingly introduces the term mathematical logic 11)
reading it as logic of mathematics.
1) The title of W. S. Jevon’s paper is characteristic: Pure Logic or the Logic of Quality part
from Quantity.
2) Yet, some “purification” of formal logic did take place. The various interpretations of
Boole-Schröder’s algebra show (today?) that the logic of concepts and one-place
propositional functions is founded by the logic of propositions. (Since this algebra
eventually remained in the background of the logical researches and was not being
frequently connected with the later logic of propositions and propositional functions, it is
only recently that this important result is used to prove in a simple way the decidability of
monadic logic in a student textbook (cf. [Q] ch.24).)
3) Cf. [D1]
4) Cf. [D2]
5) G. Peano has remarkably pointed to the trickiness of the intuitive comprehension of the
basic mathematical entities, constructing a curve which fills a two-dimensional area, [P3].
6) Cf. [P1] and [P2].
7) It seems as if this procedure were not essentially different from Boole’s, doesn’t it? Yet, it
is. Bolle’s logic is the applied pure mathematics and the former is based on the latter.
Peano’s logic is the basis of pure mathematics and here the latter is based on the former.
8) It is the beginning of Hilbert’s formalism.
9) The construction of real numbers by Dedekind’s cuts (arithmetization of continuum)
should have shown the logical definability of real numbers by rational numbers. Logical
definition of rational numbers by natural numbers is also to Dedekind’s merit, as well as
the reduction of the natural number concept to the logical concept of chain (by abstracting
the nature of its elements). The possibility of the logically founded process of abstraction
by the concept of the similarity of sets is due to Cantor. It is interesting that Dedekind’s
concept of chain (in [D2]) anticipates Peano’s axiomatic of natural numbers from [P4].
However, each of these two works point out just one of the two above stated aspects of the
critical movement in mathematics. Thus, G. Peano postulates the principle of the
mathematical induction by formally expressing it, while R. Dedekind has informally
proved it as a consequence of purely logical characteristics of chains. (Peano’s axiomatics
is not original. It can be found informally and in broader lines in Grassmann’s [G], as
quoted by Peano himself. He is one of the few mathematicians who noticed and publically
promoted the importance of Grassmann’s work, particularly in geometry.)
10) It is necessary to mention that the joining of these two basic aspects of the critical
movement (joining which we identified with the beginning of logicism) had been carried
out not only before Russell’s [R1] but even before [D2] and [P4] (which point out jus one
of the two aspects). Logicism was already originated in [F1] by G. Frege, in 1879. In this
work mathematical argument was formalized in Peano’s sense of the word, but with a
throughness which had never been achieved by Peano; it was formalized in the form of the
quantification theory (for the first time); it is here that the concept of the sequence of
natural numbers was founded, by a method analogous to Dedekind’s, in the context of the
already formalized system of arguments (for the first time). It might seem strange then
that we do not start the survey of this epoch with Frege. Nevertheless, it is in keeping with
the main stream of the development of these ideas. Frege’s work met complete
misunderstanding and it remained unknown for twenty years (until B. Russell draw
attention to its significance), and it became widely acknowledged only fifty years after it
had been published. (It is interesting that in his honorary speech on the G. Frege’s
nomination for the associate professor in Jenna, E. Abbe predicted exactly such a
sequence of events, saying that [F1] will exerce considerable influence on mathematics
but not immediately, because there will hardly be a person who could have an attitude
towards such an original entanglement of ideas.) The reasons for this misunderstanding
are multifold. Frege’s work falls into Boole’s epoch. Frege did not know about Boole’s
work and could not explicitely point to the radical difference between his own (logicist)
and Boole’s approach. Besides, the title of his work was understood (at the time it wfas
published) in Booleian way, as arithmetization of logic. From this point of view, Schröder
gave a review of [F1] in [S], by comparing Boole’s and Frege’s symbolic notation and he,
naturally, concluded that Boole’s one has advantages being taken over from arithmetic,
but he wrongly claimed that the scope of Frege’s logical analysis was not wider than
Boole’s. In Venn’s review [V1], the author came to the same conclusion, and so it is no
wander that he wrote of Frege as “one of those instances of an ingenious man working out
a scheme – in this case very cumbrous one – in entire ignorance that anything of the kind
had ever been achieved before”, from [V2] p. 415. Frege could not but regret that J. Venn,
E. Schröder and others did not try to rewrite some formulae from third chapter of his book
in Boole’s notation (cf. [F2]). They would have immediately found out that it was not
possible because Boole’s logic, after being translated into the language of the theory of
quantification, coincides with the logic of one-place predicates only, while Frege’s system
includes the complete first order logic. In connection with the just mentioned never made
translation, it should be noted that P. Tannery in his review of Begriffsschrift [T] called
Frege’s replacement of the concepts of subject and predicate by that of function and
argument fruitless and unfavourable, the replacement which founded the theory of
quantification and which enabled the logic of one-place predicates to be simply extended
to many-places predicates (replacing in this way the clumsy theory of relations, which
represents a respective extension in Boole-Schröder’s tradition). Realizing what was going
on, Frege wrote an article in which he explained the main differences between his and
Boole’s system. However, the article, as well as its short version, was refused by three
magazines. The fate of [F1] was also the fate of [F2] and of the first part of [F3] (the
second part of [F3] not being published because the editor claimed the public was not
interested in it). The first part of [F3] was reviewed by G. Peano in [P5], where he wanted
to show the predominance of his own logic ove Frege’s. However, this marked the
beginning of Frege’s successful period. By his answer (in [F4]) Frege cause Peano to
acknowledge the significance of his logic and to thank him for the improvement of his
own one. Besides, due to Peano, B. Russell got to know Frege’s work, and he (realizing
that the logicism he was aiming at had already been initiated by Frege), begun (so to
speak) to advertise Frege’s significance. On the other hand, B. Russell delivered a severe
blow to the whole Frege’s programme, by the famous “Russell’s paradox” [R2].
11) In [P1] for the first time (as for as the author knows).
[B] Bynum, T. W.: Conceptual Notation and Related Articles. Oxford: Clarendon Press, 1972.
[D1] Dedekind, R.: Stetigkeit und irrationale Zehlen. Braunschweig: Vieweg, 1872.
[D2] Dedekind, R.: Was sind und was sollen die Zahlen? Braunschweig: Vieweg, 1887.
[F1] Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des
reinen Denkens. Halle: Nebert, 1879.
[F2] Frege, G.: “Uber den Zweck der Begriffsschrift”. Sitzungsberichte der Jenaischen
Gesellschaft für Medicin und Naturwissenschaft, JZN 16, 1882/3. pp. 1-10. (Translated in
[B].)
[F3] Frege, G.: Grundgesetze der Arithmetik Begriffsschriftlich abgeleitet. Jena: H. Pohle,
1893. Band I, 1903. Band II.
[F4] Frege, G.: “Lettera del sign. G. Frege all’editore”. Rivista di Matematica 6, 1896-9. pp.
53-9.
[F5] Frege, G.: Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung
uber den Begriff der Zahl. Breslau: W. Köbner, 1884.
[G] Grassmann, H.: Lehrbuch der Arithmetik. Berlin, 1861.
[P1] Peano, G.: Notations de logique mathematique. Turin, 1894.
[P2] Peano, G.: Formulaire de mathematiques. Turin, 1895-1904.
[P3] Peano, G.: “Sur une courbe qui remplit toute une aire plane”. Math. Annalen 36, 1890.
pp. 157-160.
[P4] Peano, G.: Arithmetices principia. Turin, 1889.
[P5] Peano, G.: Review of Frege’s Grundgesetze der Arithmetik I. Rivista di matematica 5,
1895. pp. 122-8.
[R1] Russell, B.: The principles of mathematics, vol. 1. Cambridge University Press, 1903.
[R2] Russell, B.: !A Letter from B. Russell to G. Frege” in: Source Book in Mathematical
Logic, ed. J. van Heijenoort. Cambridge, Massachusetts: Harvard University Press, 1967.
[S] Schroder, E.: Review of Frege’s Begriffsschrift. Zeitschrift für Mathematik und Physik 25,
1880. pp. 81-94. (Translated in [B].)
[T] Tannery, P.: Review of Frege’s Begriffsschrift. Revue philosophique 8, 1879. pp. 108-9.
(Translated in [B].)
[V1] Venn, J.: Review of Frege’s Begriffsschrift. Mind 5, 1880. p. 297. (Translated in [B].)
[V2] Venn, J.: Symbolic Logic. London: Macmillan, 1881.
[Q] Quine, W. van O.: Methods of logic. New York: Holt, 1950.