# Is replicability necessary in the production of knowledge? Discuss with reference to two areas of knowledge.

### Submitted for IB Theory of Knowledge

In this essay, I will explore the claim that replicability is necessary in the production of knowledge. I will focus on two Areas of Knowledge: the Human Sciences – in particular, Economics; and Mathematics.

First, I will define ‘the production of knowledge’ in each discipline. Bertrand Russell said of pure mathematics that it is “the class of all of the propositions of the form ‘p implies q’” (Russell, 1903, p. 2). In other words, the production of knowledge is concerned with using logical deduction to generate proofs and theorems. It would seem that generating proofs is necessarily a replicable process: the laws of logic are constant and binding; and everyone begins from the same axioms. Hence, if some mathematician shows that those laws enable us to deduce some result from the initial postulates, then another mathematician should be able to replicate the result. For example, the Pythagorean Theorem holds that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. If one accepts the axioms of Euclidean geometry, what is meant by a right-angled triangle, and so on, this theorem is always true. As such, just as Euclid was able to prove that this is the case in 300BC, the proof can be replicated today. Indeed, the ability to replicate the proof is a key part in being confident of its truth – in justifying one’s knowledge of it. If it were to somehow turn out that the proof of this theorem could no longer be replicated, then mathematicians would cease to be sure of it. They could not rely on claims of its historical validity; they would need to see and understand it, and this understanding would enable them to reproduce the logical steps taken to prove it. This emphasis on rigour and replicability is reflected and generalised in the peer review process in Mathematics. When a mathematician proposes a new theorem or proof, other mathematicians scrutinize the proof in order to ensure that it can be replicated. If it cannot – that is, if it contains logical gaps or errors – then the mathematical community will reject it. Thus, replicability plays a crucial role in ensuring the validity of mathematical knowledge.

However, such reasoning is not flawless. I will take issue with this argument by parsing more precisely the key terms used in the claim: in particular, by analysing the relationship between ‘producing’ and justifying knowledge in mathematics; and by considering the distinction between ‘replicability’ and replication. I will begin with the first of these objections: that replicability has thus far been referred to only in the context of justifying, rather than producing mathematical knowledge. But it can be observed that when mathematicians produce mathematical knowledge, they necessarily check – that is, logically justify – their work as they go: this is done not only retrospectively by others during peer review; it is also done in one’s own head as one produces it. This entails replicating each step in one’s mind and assuring oneself as to its truth. Hence, the intimate link between producing knowledge and justifying it in Mathematics, on account of its logical nature, renders this objection unproblematic. Thus, I can turn to the criticism that ‘replicability’ must be more precisely defined. This point can be demonstrated by means of a famous example: Andrew Wiles’ proof of Fermat’s Last Theorem. The conjecture (that no three positive integers, a, b and c can satisfy the equation for integer ) was first stated in 1637. It was finally solved in 1995, over the course of two papers which together are 129 pages long. The proof relies on esoteric mathematics from algebraic geometry and number theory; and on prior theorems and ideas such as Ribet’s Theorem, Frey’s Curve, and the Tainyama-Shimura-Weil conjecture (Wiles, 1995). Wiles was able to produce the proof after seven years of intense hard work; but it is unlikely that he, or anybody else, could replicate the entire proof without significant help. Even working mathematicians who have studied number theory, algebraic geometry, and representation theory, but with PhDs and specialisations too far removed from these areas remain unable to comprehend the full proof, let alone replicate it. Rather, the few experts whose specialisation is very close to this particular area can understand various component parts of it; and together, verify its veracity. In short, as mathematics becomes more advanced and specialised, it becomes much harder to replicate findings. Despite this, results proven at the frontiers of mathematical research can be considered knowledge that has been produced. I therefore conclude that although knowledge produced in mathematics must – in principle – have the quality of replicability, it is not necessary that some unit of knowledge (a proof) can be practically replicated in its entirety. Since proofs are aggregations of stepwise, logical deductions, it is only important that each such deduction is practically replicable from the accepted axioms and laws of logic.

The criticisms lead to important clarifications, but it is ultimately true that replicability is necessary in the production of knowledge – in Mathematics. I will now turn to considering whether the same holds in Economics. Any differences here will stem from the contrasting meaning of ‘the production of knowledge’ in these fields. Economics is a Human Science: this means that it is the study of how people behave, and in particular, the mechanisms by which individuals and societies make decisions about how to allocate scarce resources, and how these decisions interact in the market (Blink & Dorton, 2020, pp. 4-6). So, although economists may seek to parameterise human behaviour – assuming individuals to be perfectly rational, selfish agents, and so on – economic theory cannot operate purely by deductions from these premises, because they are untrue in the real world. Instead, the production of knowledge must focus more on empirical observation and data than on pure logic. Replicability is almost impossible to attain in Economics, because it is not so logically certain as Mathematics. Consider, for instance, the macroeconomic theory developed by John Maynard Keynes that explained how government intervention could be used to stabilise the economy. In studying Economics, I learned of his argument that the economy is not always self-correcting (contrary to the conventional wisdom of his day) and that low aggregate demand can lead to persistent unemployment and low economic growth; and that government intervention such as manipulation of government expenditure and taxation should be used in recessions to stimulate aggregate demand and hence stabilise the economy (Keynes, 1936). But the economy of the 1930s, when Keynes was writing, was fundamentally different from the economy of today. Among other things, the world economy was far less integrated, as the world recovered from the economic impacts of the World War One. Hence any attempt to replicate Keynes’ work would necessarily be done in a set of drastically different circumstances.

It might nonetheless be argued that some element of replicability is needed if an economic result is to be widely accepted and considered valid. However, there are two core components to economic theorising: the first is chains of deductive reasoning which make use of a classical assumption: “ceteris paribus” meaning “with all other things remaining equal”; and the second is testing these hypotheses using empirically obtained data (Welker, 2018, p. 10). Take another example that I have studied in Economics: the claim that implementing a national minimum wage leads to higher rates of unemployment. Economic theory suggests that the implementation of a national minimum wage, assuming ceteris paribus, disincentivises businesses from hiring lower-skilled workers as they impose a cost to their business that they are unwilling or unable to pay; and therefore fewer such workers can find jobs, causing greater unemployment (Maley & Welker, 2022, pp. 143-144). When economists test this view, they collect data from several countries in which this has been tried, to determine the effects of the policy in general. If the results of such studies are replicable, and consistently yield that same result, then that lends credence to the conclusions of the reasoning. If instead, the study showing that the claim is true cannot be replicated, then the reliability of the study and reasoning is called into question: it is difficult to determine if the relationship is one of cause and effect, or if the result was random chance. The problem is that economists assume all other things to be equal, when they so seldom are. The economy is such a complex and dynamic system influenced by such a wide range of factors that controlling for all of these is impossible. Consider, for example, the relationship between inflation and unemployment. The Philips curve suggests that assuming ceteris paribus, there is an inverse relationship between inflation and unemployment – one increases as the other decreases. However, the American economy in the 1970s saw “stagflation”, a phenomenon in the economic cycle caused by a decrease in productivity, characterised by slow growth and high unemployment (Maley & Welker, 2022, pp. 323-329). This is not a disproof of the Philips curve generally; it is still widely accepted by economists, but it was not true when the level of productivity was not held constant. These examples therefore show that replicability is often not possible or ‘necessary’ in Economics, but simply desirable. What is often more important is the quality of the underlying research, the strength of reasoning and analysis, and the relevance and value of the insights the theory provides.

Hence, this essay has shown that replicability is necessary in Mathematics, because of its deductive, proof-based nature; the replicability of each logical step is assured. In Economics, replicability is useful in establishing reliability and validity, but it is far less vital; so it is not necessary in the production of knowledge in the Human Sciences.

# Bibliography

Blink, J., & Dorton, I. (2020).

Economics Course Companion.Oxford: Oxford Univeristy Press.Keynes, J. M. (1936).

The General Theory of Employment, Interest, and Money.Palgrave Macmillan.Maley, S., & Welker, J. (2022).

Economics for the IB Diploma.London: Pearson.Russell, B. (1903).

The Principles of Mathematics.Cambridge University Press.Welker, J. (2018).

IB Economics SL & HL.The Economics Classroom .Wiles, A. (1995). Modular Elliptic Curves and Fermat's Last Theorem.

Annals of Mathematics.