Paul Ernest argues that mathematical truths are invented not discovered and shows why 1 + 1 does not always add up to 2.
Is science a rational description of the world converging on the truth, or is it a socially constructed account? This is the realist vs relativist debate currently raging. What has gone unnoticed in this debate is that there is a parallel dispute over whether mathematics is discovered or invented.
The absolutist view of mathematics sees it as universal and objective, with mathematical truths being discovered through the intuition of the mathematician and then established by proof. The absolutists - most mathematicians - claim that mathematics must be woven into the very fabric of the world, for since it is a pure endeavour removed from everyday experience, how else could it describe so perfectly the patterns found in nature?
The opposing view, often called fallibilist, sees mathematics as an incomplete and everlasting work-in-progress. It is corrigible, revisable, changing, with new mathematical truths being invented, rather than discovered.
So who are the fallibilists? An early one was the philosopher Wittgenstein, who, in later works such as Remarks on the Foundations of Mathematics claimed that mathematics consists of a motley of overlapping and interlocking language games. These are not games in the trivial sense, but the rule-governed traditional practices of mathematicians, providing meanings for mathematical symbolism and ideas.
Imre Lakatos, another fallibilist, argues that the history of the subject must always be given pride of place in any philosophical account. His Proofs and Refutations traces the historical development of a result in topology, the Euler Relation, concerning the number of faces (F), edges (E) and vertices (V) of mathematical solids. For simple flat-sided solids, the relationship is F+V=E+2. However, proving this fact took over 100 years as the definitions of mathematical solids, faces, edges and vertices were refined and tightened up, and as different proofs were invented, published, shown to have loopholes, and modified. Lakatos argues that this shows no definitions of proofs in mathematics are ever absolutely final and beyond revision.
Philip Kitcher offers a further refinement of fallibilism in his book The Nature of Mathematical Knowledge. He argues that much mathematical knowledge is accepted on the authority of the mathematician, and not based on rational proof. Since the informal and tacit knowledge of mathematics of each generation varies, mathematical proof cannot be described as absolute.
I believe that not only is mathematics fallible, but that it is created by groups of persons who must both formulate and critique new knowledge in a formal "conversation" before it counts as accepted mathematics. Knowledge creation is part of a larger overall cycle in which mathematical knowledge is presented to learners in teaching and testing "conversations" in schools and universities, before they themselves can become mathematicians and participate in the creation of new knowledge.
This perspective offers a middle path between the horns of the traditional dilemma of whether knowledge is objective or subjective.
Unfortunately, fallibilism is too often caricatured by opponents as claiming that since mathematics is not absolutely necessary it is arbitrary or whimsical; that a relativist mathematics, by relinquishing absolutism, amounts to "anything goes" or "anybody's opinion in mathematicians is as good as anybody else's".
No fallibilist I know would subscribe to these caricaturist claims. Fallibilism does not mean that some or all of mathematics may be false (although Godel's incompleteness results mean that we cannot eliminate the possibility that mathematics may generate a contradiction). Instead, fallibilists deny that there is such a thing as absolute truth, which explains why mathematics cannot attain it. For example, 1+1=2 is not absolutely true, although it is true under the normal interpretation of arithmetic. However, in the systems of Boolean algebra or Base 2 modular arithmetic 1+1=1 and 1+1=0 are true, respectively. Thus truths in mathematics are never absolute, but must always be understood as relative to a background system. So an assumption like Euclid's Parallel Postulate and its denial can both be true, but in different mathematical interpretations (in the systems of Euclidian and non-Euclidian geometries). Mathematicians are all the time inventing new imagined worlds without needing to discard or reject the old ones.
Roger Penrose asks whether the objects and truths of mathematics are "mere arbitrary constructions of the human mind?". His answer is in the negative - he concludes that mathematics is already there, to be discovered, not invented. Plausible as this seems at first, it is argued on mistaken grounds. Mathematicians such as Penrose often contrast necessity with arbitrariness, and argue that if relativist mathematics has no absolute necessity and essential characteristics to it, then it must be arbitrary. Consequently, they argue, anything goes in mathematics. However, as the philosopher Richard Rorty has made clear, contingency, not arbitrariness, is the opposite of necessity. Since to be arbitrary is to be determined by chance rather than reason, the opposite of this notion is that of being selected or chosen. I wish to argue that mathematical knowledge is based on contingency, due to its historical development and the inevitable impact of external forces on the resourcing and direction of mathematics, but it is also based on the deliberate choices of mathematicians, elaborated through extensive reasoning. Both contingency and choice are at work in mathematics, which follows by logical necessity from its assumptions and adopted rules of reasoning, just as moves do in the game of chess. This does not contradict fallibilism - for none of the rules of reasoning in mathematics is itself absolute. Mathematics consists of language games with deeply entrenched rules and patterns that are very stable and enduring, but which always remain open to the possibility of change, and in the long term, do change.
This position weakens the criticism from absolutists that an invented mathematics must be based on whims, and that the social forces moulding mathematics mean it can be reshaped accorded to the prevailing ideology of the day. The fallibilist view is more subtle and accepts that social forces do partly mould mathematics. However, there is also a largely autonomous internal momentum at work in mathematics, in terms of the problems to be solved and the concepts and methods to be applied. The argument is that these are the products of tradition, not of some externally imposed necessity. Some of the external forces working on mathematics are the applied problems that need to be solved, which have had an impact on mathematics right from the beginning.
Originally written arithmetic was first developed to support taxation and commerce in Egypt, Mesopotamia, India and China. Trigonometry and spherical geometry were developed to aid astronomy and navigational needs. Later mechanics (and calculus) were developed to improve ballistics and military science. Statistics was initially developed to support insurance needs, to compute actuarial tables, and subsequently extended for agricultural, biological and medical purposes. Most recently, modern computational mathematics was developed to support the needs of the military, in cryptography, and then missile guidance and information systems. These examples illustrate how whole branches of mathematics have developed out of the impetus given by external needs and resources, and only afterwards maintained this momentum by systemising methods and pursuing internal problems.
This historical view of fallibilism also partly answers the challenge that John Barrow, professor of astronomy at Sussex University, issues to "inventionism". He asks how, if mathematics is invented, can it account for the amazing effectiveness of pure mathematics as the language of science? But if mathematics is seen as invented in response to external problems its utility is to be expected. Since mathematics studies pure structures at ever increasing levels of abstraction, but which originate in practical problems, it is not surprising that its concepts help to organise our understanding of the world.
The controversy between those who think mathematics is discovered and those who think it is invented will run and run. I do not expect to be able to convert my opponents. However, what I have shown is that a better case can be put for mathematics being invented than critics allow. Just as realists often caricature relativist views in science, so too the strengths of the fallibilist views are not given enough credit. For although fallibilists believe that mathematics has a contingent, fallible and historically shifting character, they also argue that mathematical knowledge is to a large extent necessary, stable and autonomous. Once humans have invented something by laying down the rules for its existence, like chess, the theory of numbers, or the Mandelbrot set, the implications and patterns that emerge from the underlying constellation of rules may continue to surprise us. But this does not change the fact that we invented the 'game' in the first place. It just shows what a rich invention it was. As the great 18th-century philosopher Giambattista Vico said, the only truths we can know for certain are those we have invented ourselves. Mathematics is surely the greatest of such inventions.
Paul Ernest is reader in mathematics education, University of Exeter and author of the forthcoming book Social Constructivism as a Philosophy of Mathematics.