If only all things were as clear as in mathematics. Clear rules of reasoning, gap-free ar-gumentation, exactly one correct answer to every question. No room for disagreement once a proof is in place. For many of us, mathematics embodies the ideal of a contradic-tion-free “Big Picture” – an ideal that not only the natural sciences, but also all other areas of discourse with a claim to objectivity are guided by. Questions such as “Which religions belong to Germany?,” “Is security more important than freedom?” or “Are we allowed to manipulate the human genome?” are discussed in the conviction that one is looking for exactly one, namely the right answer. The pluralism debate in mathematics shows that this conviction may be wrong.
The mathematical ideal has dominated our thinking for thousands of years. True, early Mesopotamian and Egyptian mathematics developed chiefly as a consequence of prac-tical needs of civilizations, such as taxation, measurement of land areas, calculation of lunar calendars, etc. However, given its exactitude, philosophy soon attributed another role to mathematics, namely that of a paradigm for human thought and knowledge. Plato considered mathematics to be the highest form of knowledge, and as such, as differing sig-nificantly from our uncertain views of the empirical world. He believed that mathematical knowledge consists in knowledge of eternal “forms” – perfect ideas that represent ultimate reality – and he considered mathematical competence essential for the acquisition of any knowledge whatsoever. Galileo saw mathematics as the “language of the book of nature,”
Kant argued that mathematics provides essential insights into a whole genre of human judgements, Cantor found God in the transfinite numbers …1 In short, the idea that math-ematics holds the key to our understanding of the basic structures of reality has dominated philosophy for thousands of years.
In the 19th century, this picture changed dramatically. The introduction of terms with little or no physical meaning separated mathematics from the empirical world. Complex num-bers, n-dimensional spaces, abstract algebras, pathological functions and non-Euclidean geometries began to establish themselves as legitimate objects of mathematical study. As
1 Plato. The Republic. Book V. in: Plato. Complete Works, ed. John M. Cooper and D. S. Hutchinson. Indi-anapolis and Cambridge: Hackett Publishing, 1997, pp. 1077–1107. Morris Kline. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972, pp. 328–329. Imanuel Kant.
Critique of pure reason, ed. Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press, 1998, pp. 110, 175, 188–189, 202. Georg Cantor. Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
Leipzig: B. G. Teubner, 1883.
a result, “mathematics exploded into a hundred fields,” as the historian of mathematics Morris Kline put it, and this led to a profound change in the understanding of its rela-tionship to the world. The empiricists of the Vienna Circle saw mathematics simply as a useful tool for the natural sciences whose theorems say nothing about the world and are to be regarded as true solely on the basis of linguistic conventions. The idea that the key to the ultimate truths of reality is not to be found in mathematics, but only in the empirical sciences, prevailed. Mathematics had taken a deep fall, from Plato’s heaven of perfect forms into the dull toolbox of empirical science.
In the 20th century, developments in modern set theory turned this picture upside down yet again. Through groundbreaking proofs and axiomatizations, it became clear that set theory is a foundation for all mathematics. Suddenly, even the most distinct of the “hun-dred fields” of mathematics could be interpreted and tested for coherence within a unified framework. Unclear terms and structures were clarified; the basic assumptions made in different forms in different areas could be identified, and through their new unified pre-sentation, previously hidden connections between different areas of mathematics became visible. Plato’s vision had come true in a certain sense: mathematics had discovered its own foundations and, as the mathematician David Hilbert described it, had developed into an autonomous “paradise” against whose unparalleled precision and certainty every other science and discourse must measure itself.
Today, however, we know that even paradises are imperfect: countless mathematical hy-potheses cannot be decided in the set-theoretical axiom system “ZFC,” the foundation of all mathematics. And it is not even clear whether there is really only one correct answer to each of those open mathematical questions. Most notorious of them is the Continuum Hypothesis, put forward by Cantor in 1878, ranked first in 1900 on the list of the 23 most important open mathematical questions, and still unsolved today. Cantor had proved that there are infinite sets of different sizes, or more precisely, that the set of real numbers is uncountable, i. e. its cardinality is greater than that of the natural numbers. Against this background, the Continuum Hypothesis expresses the assumption that there are no in-finite sets that are larger than the set of natural numbers but smaller than the set of the reals.
Some mathematicians believe that the Continuum Hypothesis has a definite truth value, i. e. that it is either true or false. Hugh Woodin, for example, professor of mathematics and philosophy at Harvard University, is convinced of its truth.2 Since 2010 he has been working on the construction of an inner model of the set-theoretical universe, which – if successful – would be a mathematical sensation. Not only because it would confirm the truth of the Continuum Hypothesis, but also because it would draw a neat, elegant and manageable picture of the infinite hierarchy of sets and thus revolutionize our understand-ing of the inner structure of infinity.
2 W. Hugh Woodin. In Search of Ultimate-L the 19th Midrasha Mathematical Lectures. in: The Bulletin of Symbolic Logic, vol. 23, no. 1, 2017, pp. 1–109.
However, not all mathematicians support this approach. Set theorists like Stevo Todorčević of the University of Toronto are convinced that the Continuum Hypothesis is wrong.3 They prefer the “forcing” method, a mathematical technique by which an infinite number of set-theoretical universes can be constructed in order to answer open questions in them.
While this technique can be used to show that the Continuum Hypothesis is wrong, from a philosophical perspective, forcing has a purely pragmatic value, as it does not promote our understanding of the inner structure of infinity.
Thus, contrary to the widespread opinion that in mathematics there is exactly one correct answer to every question and thus no room for dissent, there is de facto disagreement not only about the best methods of proof for set-theoretical conjectures, but also about how we should understand some of the most basic mathematical terms (such as “set” or
“infinity”) at all. Some mathematicians even feel reminded of “religious” or “schismatic”
controversies.4
In contrast to religion (or even politics), however, mathematicians are fairly relaxed about the disagreements they face. No set theorist would lapse into hysteria or wild insults just because a colleague defends the complete opposite of what s*he himself believes. And this despite the fact that the most fundamental truths of perhaps the most fundamental science are at stake. On the contrary, the mathematical avant-garde is currently even arguing for finally abandoning the ideal of a uniform, contradiction-free Big Picture, in favor of facing up to the pluralism reflected in mathematical practice.
One pioneer of this approach is the logician Joel Hamkins, Professor of Logic at the Fac-ulty of Philosophy at the University of Oxford. He suggests abandoning the hunt for one uniquely correct axiom system that will determinately and unambiguously answer all open questions.5 Instead, he calls for acceptance of the fact that mathematics is not a universe but a multiverse consisting of infinitely many, sometimes very different subuniverses, and that the decision which of these subuniverses an individual mathematician should investigate is made from a purely pragmatic point of view. Admittedly, this approach implies a drastic change in our view of what we call “mathematical reality.” However, it does reflect a de facto already existing situation.
If even mathematics, the paradise of exactitude and certainty, can endure such disagree-ments, should we not also strive for equal composure in discourses dealing with far less
3 Stevo Todorčević. Generic absoluteness and the continuum. in: Mathematical Research Letters, vol. 9, no. 4, 2002, pp. 465–571. Stevo Todorčević. Collapsing ω2 with semi-proper forcing. in: Archive for Mathematical Logic, vol. 57, no. 1–2, 2018, pp. 185–94.
4 Geoffrey Hellman and John Bell. Pluralism and the Foundations of Mathematics. in: Scientific Pluralism (Minnesota Studies in the Philosophy of Science, vol. 19), ed. Stephen Kellert, Hellen Longino and Kenneth Waters, Minneapolis: University of Minnesota Press, 2006, pp. 64–79.
5 Joel David Hamkins. The Set-Theoretic Multiverse. in: The Review of Symbolic Logic, vol. 5, no. 3, Sep-tember 2012, pp. 416–449.
precise matters? The pluralism debate in mathematics shows how important it is to be able to withstand tensions even over long periods of time. In extreme cases, it can take centuries to find the right answer to a difficult question. And sometimes the right answer does not even arise from theoretical considerations, but from what proves itself useful in practice.
Perhaps, then, it is best to approach seemingly unsolvable questions, even if they touch on the most fundamental aspects of human existence, pragmatically rather than dogmatically.
A German version of this text was published in Süddeutsche Zeitung on January 7 2019, p. 9.