A popular Indian fable tells of a Brahmin by the name of Lahur Sessa, who travels south to seek an audience with King Ladava at the royal palace. Received by the king, Sessa presents him with his new game of Chaturanga — the game which we now know as chess. Delighted by the gift, the king offers his guest anything of his choosing from the royal treasury. But the Brahmin instead asks that he be paid in grains of wheat, one grain for the first square on the board, two for the second, four for the third, eight for the fourth, and so on, until the sixty-fourth square. Disarmed by such a meagre request, King Ladava immediately agrees. However, he subsequently learns that the quantity asked would be greater than all of the grain in the kingdom. Humbled by the lesson, the king appoints Sessa his vizier.
Mathematics is unique amongst scholastic pursuits because, given the correct assumptions, it is able to make predictions about the world that can be known with absolute certainty, and without the reliance on empiricism. We do not need to count the grains on the chessboard from the fable, for example, to know that there would be over eighteen quintillion (specifically 18,446,744,073,709,551,615) by the final square — the equivalent of around one trillion metric tonnes of grain. Through the application of mathematical logic, we can determine that fact beyond any doubt.
It is important, before proceeding, to distinguish between two related terms that in everyday English are often used interchangeably, but which, in this context, will communicate distinct ideas.
Rationality combines logic and reason to draw conclusions about probable truths, in such cases that there is insufficient information to make a prediction with certainty. Thus, a thoroughly rational conclusion can be imperfect, or even wrong, just as an entirely irrational conclusion can prove to be correct. Furthermore, when the influences of known and unknown quantities are comparable, a rational theory can still be rationally disputed. The validity of a particular rationale depends upon the strength of the evidence supporting it.
We should understand, therefore, that the merit of rationality exists in the probability of its predicting the truth. The more rational the argument, the more likely that argument will describe the reality.
When Danish astronomer Tycho Brahe rejected the theories of his German contemporary Johannes Kepler, theories which were founded upon acceptance of the thitherto unproven heliocentric astronomical model of Nicolaus Copernicus, he did so out of rationality. His own astronomical measurements, the most rigorous and accurate of the time, were nevertheless too imprecise to detect parallax shift. And so, in the absence of empirical data that would support the Copernican model, he proposed an alternative geocentric one that his data would substantiate, in which the other planets, only, orbited the sun. Kepler was ultimately proven to be correct, but Brahe might fairly be recognised as having been the more rational scientist.
Logic, by contrast, describes an undeniable and immutable relationship that exists between two or more things. If a proposition contains a flaw in logic, the failing is not intrinsic to the logic itself, but rather to its application. Similarly, the application of logic to a proposition might be faultless, but the assumptions upon which that proposition is founded may still be erroneous. The utility of logic is therefore only as great as the accuracy of its inputs.
In the history of mathematics, there have been numerous instances in which mathematical proofs were proposed and accepted, but later invalidated. Two such examples were the fallacious proofs of the Four-Colour Theorem, tendered independently by Alfred Kempe and Peter Guthrie Tait, in 1879 and 1880 respectively. Each was accepted for a period of a decade before being discredited on the bases of logical errors. It was not until 1976 that Kenneth Appel and Wolfgang Haken, employing computational methods developed by Heinrich Heesch, provided a proof that, despite requiring later corrections, eventually earned broad acceptance. Since it remains infeasible to check the logic of the proof manually, however, philosophical debate still surrounds its validity. Notwithstanding, some universal truth of the problem exists — that is, the theorem is either correct or incorrect — and only logic can determine that truth with absolute certainty.
For those of us who are not mathematicians, perhaps a far simpler, algebraic proof might better serve to illustrate the permanence of logic. Consider the following:
Presented with such a contradiction, we can conclude either that mathematics is inherently imperfect, or alternatively, that we have erred in our application of mathematical logic. The more philosophical amongst us might be tempted to consider the former, but we would do well to exercise caution in drawing that conclusion. Millennia of history have given us little reason to question the certainty of mathematics. When it fails, it does so because our derisory minds are incapable of recognising its patterns or grasping its logic. Two is certainly not equal to one, and mathematics has no difficulty in discerning that fact.
(I will leave the contradiction above unexplained, hopeful that the reader will find the flaw in mathematical logic unaided.)
Our two terms, therefore, are distinguished by the fact that whilst logic alone may be applied when we have sufficient information to draw a certain conclusion, rationality must be applied when our information is insufficient. Logic is absolute, whilst rationality appeals to probability.
No single definition can describe the field of mathematics perfectly, but the Concise Oxford Dictionary of Mathematics (5th edition) perhaps defines it most adequately as “the branch of human enquiry involving the study of numbers, quantities, data, shape and space and their relationships, especially their generalizations and abstractions and their application to situations in the real world.” Crucially, those generalisations and abstractions are only valid because they are founded in logic. Indeed, mathematics is logic, as it is applied to certain branches of enquiry. But that same logic can be applied to anything. Herein lies the importance of mathematics.
Lamentably, influenced by a century of dogmatic Western educational progressivism, the true value of mathematics is seldom appreciated by contemporary teachers and educators. This is why more than half of our students report that they hate, dislike, or are indifferent to the subject; it is why Asian countries occupy the first seven rankings in mathematics in the Program for International Student Assessment; and it is why human discourse, at all levels of society and education, is discordant, dogmatic, and replete with logical fallacy, to which most of us appear to be blind.
In virtually every popular scientific, political, or philosophical debate, the primary point of separation between its antagonists is not the facts themselves, but rather rhetorical fallacies constructed around interpretation of those facts by one party or the other. The fallacies can easily be described logically, and such clarification should result in consensus between the parties. Yet, we are invariably imperceptive or resistant to the logic, and hence to change.
Consider the syllogism: All living things need water. Fish need water. Therefore, fish are living things. According to the work of Canadian psychologist Keith Stanovich, 70% of university students would accept the validity of this conclusion. But they would be wrong. Whilst fish certainly are living things, we cannot reach that conclusion from the premises given. (The first premise does not state that all that needs water is living.) Nevertheless, since the conclusion seems reasonable, and since it is consistent with our understanding of the world, we are tempted to accept the truth of its logic.
In his 2011 book Thinking, Fast and Slow, Israeli psychologist Daniel Kahneman defines two distinct systems of thought. Fast thinking is instinctive and heuristic, and consequently subject to emotion and cognitive biases. Its primary function is decision-making efficiency. By contrast, slow thinking is more deliberative and rational. Its function is to solve complex problems involving abstract and relational stimuli. Whilst it is still limited by the cognitive ability of the individual, slow thinking is more likely to lead to a logical or rational judgement.
Mathematics, particularly when it is difficult and unfamiliar, is incompatible with fast thinking. With the correct approach, it compels us to think slowly, deliberately, and rigorously. And, founded in logic, it trains us to identify and avoid cognitive biases and logical fallacy. It further promotes objectivity, focus, patience, reflection, and clarity of organisation and communication — all interdisciplinary cognitive skills.
The educational establishment does appear to recognise the importance of mathematics education; the subject certainly remains a major feature of almost every school curriculum. However, even a cursory examination of the arguments surrounding the scope and breadth of its inclusion demonstrates that teachers and educators, from all sides of the argument, are fixated on the utility of its direct application. We have long since come to view mathematics education, as with all education, as entirely utilitarian and synonymous with vocational training. That is, mathematics is taught solely in order to provide students opportunities for work in the fields of science, mathematics, engineering, and technology.
But if that is our only motivation, most students have little reason to study mathematics past primary school. In the United States, for example, the Bureau of Labor Statistics estimates that technical occupations represent only 6.2% of the total workforce (2015 data). Furthermore, three quarters of technical graduates do not work in technical fields. Even accounting for the predicted growth of these sectors, nine out of ten students will never employ anything more advanced than whole numbers, fractions, ratios, percentages, rates, roots, exponents, and arithmetic — all topics that children are routinely taught by the age of ten. Furthermore, since electronic calculators are now a fixture of classrooms as early as primary school, students’ understanding of those topics need amount to little more than definitions.
Detractors of mathematics further propose that it is of limited importance, since it has, at times, been wrong, or because it has hitherto failed to solve one problem or another, or because computers can now perform calculations orders of magnitude more quickly, and with greater accuracy, than human beings. However, such criticisms confound mathematical logic with rationality, mathematics with mathematicians, and education with application, respectively.
As we have seen, a conclusion can be rational yet incorrect. Beyond the beautiful world of pure (abstract) mathematics, applied mathematics necessarily works with inputs derived from real-world empiricism. And observations can be inadequate, exceptional, or erroneous. Consequently, it is a matter of statistical certainty that some rational, scientific conclusions will ultimately prove to be wrong. Similarly, mathematicians are human, and occasionally prone to making mistakes. But this does not imply that irrational, arbitrary, or instinctive thinking is somehow more reliable or meaningful — it is demonstrably less so — only that, just as we can err in our application of logic, so too can that logic be limited by our knowledge of the problem. The combination of empiricism, logic, and rationality remains by far the most reliable predictor of world realities.
However, the greatest value of mathematics education exists not in its direct utility, but in the fundamental cognitive abilities that it teaches and develops — abilities which we, as a society, are desperately lacking, and in whose absence our discourse and actions are squandered, manipulated, and abused. In this ostensibly democratic world, in which we are bombarded with misinformation, propaganda, and dogma, (it is not an overstatement to say) these skills are becoming increasingly imperative to our very survival.