The rise of Mathematical Logic: from Demonstration to Laws of Thought to Foundations for Mathematics

Revised Nov 2022, Jan 2023

In this article we look at the evolution of logic from its earliest form in the demonstration of truths to the rapid development of mathematical logic in the 1800s at the start of the “golden century” of logic (1850-1950). We also look at the rise and surprising dashing of hopes for the formalist program.

Chronology of Logic as a Mathematical Program

This short chronology takes the reader through the context behind the formalist program in logic, why it meant so much to mathematicians, why it ultimately failed despite such a strong sense that it should succeed, and what further discoveries by Paul J. Cohen and others have discovered about the pure world of thought and the possibility of alternative mathematical universes, much as the 1800s discovered alternative geometric universes.

1. For most of its history, logic was a rhetorical subject.
2. For Aristotle, logic was primarily concerned with syllogisms, which explored correct/incorrect inferences using notions of predicate logic (there exist, for all) combined with semantics (meanings of terms) and sets (classes and membership).
3. Propositional logic (and, or, not, imply, etc.) was a lesser known logic of the Stoics.
4. Leibniz, in the 1600s, was fascinated with the possibility of a purely conceptual reasoning system, once which could dispense of obscuring rhetoric and operate at the conceptual level using ideograms. This was his calculus rationatur, or universal calculus, something which animated his work in mathematics and his continuing search for the fundamental concepts underlying mathematical insight/reasoning/truth. (see reference)
5. In the 1800s, progress in abstract algebra showed the need for value of abstract symbolic representations separating symbols from interpretation, were starting to become clearer (see George Peacock: How Algebra became abstract), George Boole showed that logical operations on Propositions could be mapped into an algebraic system (Propositional or Statement Logic). (see Cohen/2002)
6. In analysis, Fourier in the 1800s showed that every function, even a square function, could be resolve into an infinite series of oscillations, much to the surprise, and uneasiness of many.
7. Cauchy, Weierstrass, and Bolzano, were all involved in setting contemporary analysis on a secure foundation, after results from Fourier using infinite series, caused considerable unease.
8. In geometry, Gauss, Bolyai, and one other, almost simultaneously, found that negating the parallel postulate, rather than leading to a contradiction, led instead to a consistent alternative geometry, non-euclidean geometry. It turned out that Riemann and others, culminating in Einstein, showed that in fact this alternative geometry was the fabric of the universe in General Relativity and the curvature of space-time, and that the hitherto so certain Euclidean geometry was in fact the simple geometry of presumed flat spaces.
9. Hilbert, taking apart the logical rigour of Euclid, rebuilt Euclidean geometry with the logical gaps filled in. (Recall, Hoyrup showed that Euclidean geometry was in fact a 200 year enterprise by the Greek philosophical school of putting the 2000 year old practitioner’s geometry of the Babylonians and before them of the Sumerian/Akkadians onto a proper logical foundation.)
10. Separately, Cantor was exploring set theory as a way to set mathematical reasoning on foundation based on properties of objects, or classes, and discovered a great many surprising facts as a result of his new found precision. (see Cohen/2002)
11. Dedekind, working with Cantor, used the set theory to put the real numbers on a secure foundation with his association of a real number with a cut, i.e. each real number is in fact the limit of a one-dimensional subset.
12. In the early 1900s, Peano scoured mathematics, attempting to find symbolical representations for recurring concepts. Single-handedly after Francois Viete in the 1500s, Peano invented many of the symbols of modern mathematics (see reference). This was a major step forward in the Leibnizian program of purely conceptual reasoning.
13. Frege scoured mathematics looking for the primitive elements or building blocks for all mathematical thought. His work was a major step forward in a reductionist program for mathematics. Frege showed that much of mathematics could be represented using the newly created set theory of Cantor, propositional logic of Boole, and a new predicate logic using the symbols for all and there exists. (see Cohen/2002)
14. Bertrand Russell, enamoured by the precision of Peano and Frege, embarked on a mission to reduce all of mathematics to formal principles, thereby attempting to secure for mathematics the sufficiency of the formalist hypothesis: i.e. that mathematics could be reduced to logic, i.e. from logic, set theory could be constructed, and using logic and set theory, all of mathematics could be built up and justified. But he found fundamental contradictions could be generated through a naive formulation of set theory based on the use of properties as defining sets.
15. Zermelo and then Fraenkel tried to provide an axiomatic system to rescue set theory from the contradictions that could be formed using the property formulation of sets in an unrestricted way.
16. The issue bothering mathematicians like Hilbert was the fact that axiomatic systems could be set up without any assurance that they were consistent, i.e. they could produce contradictions, A and not-A. Equally, one would like to be assured that with a precise language, in a consistent system, like logic or set theory, whatever could be formulated was decidable in the system, i.e. had a definite truth value, either true, or false. This would have ensured that all of mathematics could be represented as a formalist endeavour.
17. But Godel brought this aspiration to a shuddering halt with this two incompleteness theorems and some relative consistency proofs.
18. Cohen, in 1963, showed that two major axioms were actually independent of the ZF axioms, namely the axiom of choice (AC) and Cantor’s continuum hypothesis (CH), as well as several relative consistency proofs (if ZF is consistent then so is ZFC, and if ZFC is consistent, then so is ZFC+CH). But Cohen equally showed that consistent models could be created in which we had ZF and not-AC or also ZF and not-CH. This dropped the final psychological bomb on a 100 year old program, i.e. that in set theory, just as had happened in geometry for 2000 years, actually, the model we think we work in is just one of a number of possibilities, and that the axioms that we intuitively hold, are just that, our assertions of the kind of reality in which we want to work, but they are not foregone conclusions, and there can be equally consistent set theories in which one or both are false.
19. Feferman, in an address to the AMS, raised the question: Does Mathematics Need New Axioms? Cohen also spoke of the search for new axioms. Godel also spoke of the search for new axioms.

Period-based Explorations

We’ll now go back to the earliest beginnings and look more closely at the major periods of development in Logic.

Early Demonstration/Logic: from 2600 BCE to 1800 CE

The logic of demonstration has been present in mathematics since the rise of the first mathematical scribal schools in ancient Sumeria c.2600 BCE [Hoyrup, 2017]. Two thousand years later, the ancient Greeks, beginning with Thales and Pythagoras c550 BCE and continuing over the next 200 years, developed a rigorous deductive method for geometry built on undefined notions, axioms, and postulates. Postulates required demonstration (proof) using previously proved postulates or by reference to the axioms or undefined notions. By 300 BCE, this tradition of ancient Greek mathematics had culminated in the magnus opus of geometry, Euclid’s Elements, which stood for the next c.2000 years as the pinnacle of logical deduction and the paragon of a logical system. In philosophy, the ancient Greeks explored human reasoning, identified fallacious reasoning and clarified the principles of valid reasoning. This tradition culminated in the works of Aristotle (c.350 BCE). In the mid 1600s, Leibniz renewed logical investigation with his inquiry into the formalization of logical arguments, or a logical calculus, through the search for a universal characteristic [Ariew/Garber/1989]. But it was not until the mid-1800s, prompted by various crises in algebra, geometry, and analysis, that logic and logical systems received a sustained re-examination that led to the reformulation of the subject using mathematical methods, the so-called mathematical logic, which ultimately found a place, with set theory, at the foundation of mathematical argument. Today, like in geometry, we recognize not just one Logic, but many self-consistent logical systems, of which the most common are:

1. propositional logic (the logic of propositions and their combination using propositional connectives), and
2. predicate logic (quantified propositions, e.g. some, all, none, at least one, including syllogisms),
3. inductive reasoning (extrapolation to a general principle based on the evidence of specific cases, based upon the well-ordering of the integers and the principle of mathematical induction),
4. multi-valued logic (non-binary logics e.g. fuzzy logic),
5. modal logic (including the logic of possible and necessary),
6. various other non-standard logics.

The Development of Mathematical Logic

After Aristotle (384-322 BCE), the next figure whose vision for logic most foreshadowed developments in mathematics, computer science, and mathematical logic, was Gottfried Leibniz (1646-1716), a contemporary of Newton (1643-1727) and independent developer of the differential and integral calculus. Leibniz’s program for logic was to develop an essentially symbolic, arithmeticized, mathematical language that could automate the reasoning process, making it clear which arguments were valid and which were not. [Ariew/Garber, 1989, pp.5-10] His intent was to arithmetize philosophy, i.e. reduce arguments to symbols with a mechanical calculus guaranteed to produce only valid results. This sweeping project, he writes that he conceived at age 14, and which he then expressed in its pre-mathematical stage in his dissertation on The Art of Combinations (1666, aged 19). [Rutherford/2012,Ch.2,p.1] Following his move to Paris, his exposure via Christian Huygens to contemporary mathematics and science, and his success prototyping a mechanical calculator capable of the 4 arithmetic operations and the extraction of roots (i.e. an improvement on the calculator of Pascal), his approach crystallized into developing a universal characteristic language (1678-79, at 32-33 years old). This search occupied the rest of his life, and is present in all his mathematical innovations, including the invention of the differential notation which he used to explicate the calculus. Similarly, Leibniz imagined an arithmetic for spatial geometry presaging what would later become the development of the vector calculus. [Crowe/1967] While these thoughts on Logic, when gathered to gather, are prescient of subsequent logical developments, so disorganized and diffuse were his papers, notes, and letters, that it took c.250 years for his Nachlass to be catalogued (1895), and thereby to begin to influence logical thinking (beginning with Louis Couturat who published the groundbreaking volume The Logic of Leibniz (1901) which was the first comprehensive presentation of Leibniz’s logical ideas in a single place.

This desire for the arithmetization of reasoning became a reality with the rise of logical symbolism and logical calculi for mathematics that was developed at the end of the 1800s, primarily at the hands of Guiseppe Peano (1858-1932) and Gottfried Frege. It led ultimately to the development of binary two-valued logical foundations upon which have been developed mechanical methods in modern logic and the theory of computation and the digital computer.

The Mathematical Impetus for a Re-Examination of Logic

During the second half of the 19th century (late 1800s), a number of mathematical problems and questions were raised, some of which had withstood solution for over 1500 years since the time of the Greek geometers. The solutions that were found raised more questions than were answered, and prompted investigations into the logical foundations of mathematics.

The first motivation was from geometry, where the demonstration of two entirely consistent non-Euclidean geometries — the hyperbolic and elliptic geometries — raised troubling questions about the sanctity of the classical Euclidean geometry that had been assumed to be the only possible geometry. In particular, it was the manner in which these alternate geometries were obtained that was troubling. Each of these geometries had been obtained by entertaining variations of the parallel postule and attempting (fruitlessly) to derive a contradiction. Thus, although Euclidean geometry was a clearly satisfactory model of geometry for much of the physics of the time, the fact that these non-Euclidean geometries were non-contradictory logical constructions was shocking, even more so because it was believed that they were logically sound but without models in nature. That they were contradiction free suggested that the axioms of Euclidean geometry might not be so self-evident after all.

In applied geometry, the stunning successes of general relativity and its essential use of non-Euclidean geometry (the metric tensor and Riemmanian geometry) suggested strongly that what had formerly been taken as absolute truth (the axioms of geometry) were in fact the basis for a particular model of the concepts of geometry: distance, angle, measure, space, and so on, but that equally applicable models could require quite different geometries for their structures to be understod. This forced an appreciation of axiomatic systems to be considered as models with axioms as assumptions for the model, rather than as self-evident first principles, which was their original notion.

A second motivation for foundational inquiry was the consequence of over a century of progressively increasing attempts to set the Calculus and, more urgently, the Fourier Analysis, on a rigorous, contradiction free foundation. The business of infinite summation, convergence, and the approximation of the mathematical functions of physics by infinite sums of functions led urgently to questions about the nature of the real numbers, and led Georg Cantor (1845-1918) to develop the set theory and an heirarchy of infinities, neither of which were well received by the broader mathematical community.

A third impetus for work in logic was the momentum building in the exploration of various abstract structures and their algebraization. Evariste Galois (1811-1832) had demonstrated the power of novel abstract algebraic methods (groups, splitting fields, and symmetries in permutations of roots) to triumph over thorny classical geometric that had escaped solution for almost two thousand years and knotting algebraic problems that were four hundred years old. (The two-thousand year old problems were from the ancient Greeks, namely the duplication of the cube, the trisection of an angle, and the quadrature of the circle. The four-hundred year old problem was finding a solution equation in radicals of polynomials of degree higher than four. This problem was the central question of the algebra of polynomials that had vexed mathematicians since the days of the Italian wranglers of the 1400s — namely, the search for a closed form expression for the roots of a polynomial in terms of algebraic combinations of its coefficients.)

The new methods of Galois, combined with the discovery of matrices and quaternions, touched off substantial developments in the study of abstract algebraic systems. The algebraic structure of logic and sets was investigated by C.S. Pierce (1839-1914) in America, Augustus De Morgan (1806-1871) in England, and George Boole (1815-1864) in Ireland. The discovery and elaboration of the algebraic structure of propositional logic was one of the areas in which modern notions moved logic into new directions — the notion of a boolean algebra.

A fourth area of activity that set the stage for advance of logic were the investigations into the foundations of mathematics by Peano (1858-1932), Gottlob Frege (1848-1925), Moritz Pasch (1843-1930), and Bertrand Russell (1872-1970) in the waning years of the 1800s and the start of the 1900s. Peano and Frege’s effort enhanced the precision of mathematics through the use of notation that could apply generally to all of its various problems. Russell’s participation actively pursued his belief in the possibility of reducing mathematics to logic, something which neither Frege nor Peano believed would be successful. [Kennedy/2002] This effort was eventually proved impossible by the fundamental impossibility results of Kurt Godel (1938-1978) in the 1930s.

A fifth impetus was the push toward formalism from internationally respected mathematician David Hilbert (1862-1943), made tangible in a challenge at a mathematics congress address “Problems for the 20th Century”. [Hilbert/20thCentury] The individual problems of this address would shape much of the work that went into Logic and Foundations Study in the first half of the 1900s. This was one of the most fertile periods in the history of logic and included such luminaries as Ernst Zermelo (1871-1953), Godel, Alfred Tarski (1901-1983), Alonzo Church (1903-1995), Willard Quine (1908-2000), Stephen Kleene (1909-1994), Emil Post (1897-1954) Thoralf Skolem (1867-1963), Leopold Lowenheim (1878-1957), and Paul Cohen (1934-2007).

Reflections on Mathematical Logic

There are many excellent textbooks on Mathematical Logic, so this is not the place to teach the subject. However, there are a few observations and reflections which may be worth making, to clarify a certain perspective on logic and mathematics.

It is sometimes thought by non-mathematicians that there is one true Logic (with a capital “L”). This essentialization of Logic is similar to the essentialization of Justice, Truth, Mathematics, and other matters in which it is convenient to imagine a singular immutable form whose properties are self-evident and beyond question. Geometry (with a capital “G”) was viewed in this way from Euclid to the discovery of non-Euclidean geometries, a situation which served only to stifle independent thought in the subject for almost 2000 years.

One of the aspects of modernism that has permeated almost every discipline, mathematics and science included, has been the realization that no area of human knowledge is beyond question. This led to the re-emergence of axiomatics in modern form as a philosphy of knowledge. In its modern form the axiomatization of a subject is made through distillation and extraction of its essential ideas and its arrangement into a theory built deductively from these first principles. The axioms themselves are viewed as instances of number of possible alternatives. An example is the co-existence and development of Euclidean and non-Euclidean geometries, in which the alternative formulations also found real-world models. It is the repeated experience of this — from non-Euclidean geometry, to abstract algebraic systems, to non-standard logics — that led mathematicians, among them Hilbert, to prioritize the formal elements of mathematical development in order to remove from the presentation of mathematics any parochial elements from this or that particular model, interpretation, or field of application.

From this perspective formal logic does not care how truth values are assigned to a statement, nor which of the statements get which values. What it does care about is that every statement has a well-defined truth value. In this situation, what formal logic proposes to do is to establish an axiomatic system in which valid reasoning can be explicitly described and in which every argument can be evaluated for correctness. To achieve this, a useful syntax has arisen over 150 years that allows the vagueness of natural language to be replaced by a symbolic language that is exceptionally well suited to the kind of reasoning that occurs in the narrower area of mathematical reasoning.

There are, of course, disadvantages to a highly formal style. In particular, intuition, motivations, and with them accessibility, are scrubbed out, resulting in the so-called lapidary (marble-ite) style of advanced modern mathematics. This does not mean that there is no intuition; on the contrary! However, in presentation, pains are taken to expunge from the main lines of development all logically non-essential traces, in order to be able to inspect the resulting logical system objectively and with precision, and to identify the important abstract structures underpinning the results. The wider the diversity in the specific subjects being synthesized into the general theory, the more abstract will be the presentation, requiring to maintain maximum independence from specific instances. This is the case in functional analysis and topology, in abstract algebra, modern analysis, and measure theory.

Appendix 1: Unwinding classical influence in mathematics

The advances in Algebra were:
(A1) the discovery of the cubic formula (del Ferro, Tartaglia, Cardano, early 1500s) and quartic formulas (Ferrari, 1540)
(A1) quintic formula for a solvable quintic (not all quintic), (Malfatti, 1771)
(A2) the insolvability of the quintic by radicals, Part 1 (Lagrange; Ruffini, 1799; Abel;), Part 2 (Galois, 1832), Part 3 (Betti, Serre, Weber, Artin)
(A3) the impossibility of quadrature of the circle in rational or algebraic numbers, and
(A4) the impossibility of duplication of the cube by straightedge and compass.
In Geometry they were:
(G1) the invention of coordinate geometry,
(G2) the unification of the transcendental functions with imaginary numbers in the theory of series,
(G3) the discovery of the independence of the parallel postulate, and
(G4) the creation of non-Euclidean geometries.

Sourcebook on Logic

This section holds primary sources influential in the historical development of logi.

1. Aristotle/360 BCE, the Organon is the combined set of his six writings on reasoning: Topics, On Interpretation, Prior Analytics, Posterior Analytics, Categories, On Sophistical Refutations.
2. Theophrastus, the successor of Aristotle, who strengthened proof theory, and was the first to describe the principles of implication (modus ponens) and contraposition (modus tollens). These are the 1st and 3rd of Hilbert’s 3 axioms for statement logic.
3. Leibniz/1670 works on logic (4 areas)
4. Leibniz/1685, Two Appendices by Leibniz, in [Lewis/1918], pp.373.ff
5. [Peacock/1830] Two page summary of the 30pp original preface to A Treatise on Algebra (ebook), George Peacock, 1830. The 2nd edition was completely rewritten in 2 vols separating vol 1. Arithmetical algebra (1842) from vol 2. Symbolical algebra (1845) (ebook). Reviews were still favourable 100 years later! (1942)
6. [Boole/1847] The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, George Boole, 1847, MacMillan and Co. Boole’s work would lead to a Boole-Pierce-Schroder tradition of Algebra of Logic that extended then to Tarski/1941. (Note that Schroder developed his parallel work independently of any knowledge of Peacock or Boole.)
7. [DeMorgan/1847] Formal Logic, or the Calculus of Inference, Necessary and Probable, Augustus De Morgan, 1847, Taylor & Walton
8. [Boole/1854] An Investigation into the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, George Boole, 1854, MacMillan and Co.
9. [Jevons/1864] Pure Logic (1864), and The Substitution of Similars (1869), William Stanley Jevons; Overview of work on logic, 1864
10. [Peirce/1870,1897] The Logic of Relatives (1897), The Monist, Peirce. Overview of work on logic.
11. [Peirce/1880] On the Algebra of Logic, 1880, C.S. Peirce
12. [Peirce/1885] On the Algebra of Logic: Contribution to Notation (1885)
13. Cantor/1874 work on infinite set theory
14. [Schroder/1877] work on algebra of logic (Operationskreiss), expanding Boole’s system. Subsequently, Whitehead/1898, and Lewis/1918 would base their work on his 1877 work. (Note that Schroder developed his parallel work independently of any knowledge of Peacock or Boole.)
15. Frege/1879, Conceptual Notation, 1879
16. Frege, List of Papers, in English
17. [Venn/1881] Symbolic Logic, John Venn, 1881, MacMillan & Co., refined the representation of logical operations anticipated by Leibniz (see above), Euler (Euler rings), and Hamilton (1861)
18. The first truth table was introduced by Christine Ladd-Franklin’s 1881 paper in Peirce/1883 Studies in Logic.
19. [Schroder/1890] 4-volume work on the Algebra of Logic: 1890, 1891, 1895, last volume published posthumously in 1905.
20. [Kennedy/2002] – Hubert Kennedy, Twelve Articles on Guiseppe Peano, 2002
21. [Ariew/Garber/1989] – transl. Ariew/Garber, Philosophical Essays of G.W. Leibniz, 1989, Hackett
22. [Rutherford/2012] – The Logic of Leibniz (PDFs), English Translation of Louis Couturat’s Logique de Leibniz.
23. [Hilbert/1895] His bibliography. Hilbert’s Foundations of Geometry had underlined the need for rigorous axiomatic, consistent basis for reasoning and the foundation of arithmetic to underpin the foundation of analysis (Dedekind) which in turn was needed to underpin the foundations of arithmetic. [Ewald/Sieg/2013] Hilbert’s Lectures. [Mancosu/1999]. [Sieg/2014], 1917-1922 program, [Zach/2006] Hilbert’ Program Then and Now
24. Hilbert, 1900, Problems of the 20th century
25. [Gray/Rowe/2000] The Hilbert Challenge, by Jeremy Gray and David Rowe, 2000
26. [Lewis/1918] A Survey of Symbolic Logic, with two extracts from Leibniz, C.I. Lewis, 1918
27. [Bernays/1918] Bernays/Hilbert results on completeness of propositional logic, and other areas. See [Zach/1999]
    

    References and Recommended Reading

28. Great Moments in History of Logic – 30 of the Key People who developed Modern Logic (2004)
29. A Catalog of Non-Standard Logics by Peter Suber
30. [Walicki/2006] – Walicki, A History of Logic, Part I of Introduction to Logic (PDF), 2006-2011 (book 2012), World Scientific
31. [Crowe/1967] – Michael Crowe, A History of Vector Calculus, 1967, Dover Publications
32. Jens Hoyrup, Lengths, Widths, and Surfaces: A Portrait of Old Babylonian Algebra and Its Kin, 2002, Springer

– explains how Greek mathematics put the logical rigour to previous babylonian geometric algebaic knowledge

33. Judith Grabiner, 1971?, Mathematical Truth is Time Dependent
34. [Hoyrup/2017] – Jens Hoyrup, Algebra in Cuneiform: Introduction to an Old Babylonian Geometrical Technique (PDF), 2017, Open Access Edition

Further Reading in Logic and Set Theory

35. [Cohen, 1966] – Paul J. Cohen, Set Theory and the Continuum Hypothesis, £4.30 (used)
36. [Cohen, 1963] – Paul J. Cohen, The Independence of the Continuum Hypothesis, Part I
37. Cohen & Davis, 1968, Non-Cantorian Set Theory, reprinted in Davis & Hersch, 1981
38. [Cohen, 2002] – Paul J. Cohen, The Discovery of Forcing, Rocky Mountain Journal of Mathematics
39. [Cohen, 1963] – Paul J. Cohen, The Independence of the Continuum Hypothesis, Part II
40. [Cohen, 1967] – Paul J. Cohen, Remarks on Mathematics and PhD Advisement
41. [Cohen, 2005] – Paul J. Cohen, Skolem and Pessimism about Proof in Mathematics
42. [Chow, 2001] – Timothy Chow, Forcing for Dummies (online)
43. [Chow, 2008] – Timothy Chow, Beginner’s Guide to Forcing
44. [Kanamori, 2008] – Cohen and Set Theory
45. [Hamilton/1988] – A.G. Hamilton, Logic for Mathematicians, 1988
46. [Feferman, xxxx] – Solomon Feferman, Does Mathematics Need New Axioms?
47. [Feferman, xxxx] – Number Systems, 2nd ed.
48. [Godel, xxxx] – Dover
49. [Godel, 1947] – Cantor and the Continuum Hypothesis
50. [Davis & Hersch, 1981] – The Mathematical Experience, pp.222-236, Non-Cantorian Set Theory
51. [Zadeh, 1977] – Fuzzy Logic
52. [Kreinovich, 2011] – In the Beginning was the word, and the word was Fuzzy (Online)