Mathematics, the study of numbers and shapes, is generally considered profound, but it originated from acts as simple as counting and generalising. In ancient Greece, some people were intrigued by the patterns that pebbles gave: three pebbles formed a triangle, four pebbles, a square; six, a larger triangle, and nine, a larger square (Figure 1)…
The pattern persisted regardless of what sort of pebble, or in fact whatever, they used, and by some tricks on symbols they were able to predict a triangle or square number as large as they wanted (Campbell, 2013). Over time, this pebble-counting art has developed so much that today mathematics forms the basis of all natural sciences. However, just what is the nature of mathematics? In specific, when the ancient Greeks knew how to generate a square number via calculation (i. e. symbol manipulation), did they discover an innate rule of nature, or did they merely invent a game of symbols? Is mathematics itself a discovery or an invention? In order to be clear, ‘discovery’ is defined as a known fact or system of facts existing independently of human thoughts, and ‘invention’ means otherwise. On one hand, because generally a true mathematical statement cannot simultaneously be false, it is natural to conclude that mathematics is about absolute truths. This is the idea of the Greek philosopher Plato, who stated that mathematics constituted an eternal, omnipresent, and perfect edifice, whose existence did not depend at all on whether we were there to explore (Abbott, 2013). On the other hand, unlike discovery-based sciences like physics and chemistry, mathematics works and develops almost solely through abstract thinking. Therefore, some more modern mathematicians like David Hilbert proposed that mathematics is more of a human creation like rules of chess, and exists nowhere beyond the human mind (Hilbert & Bernays, 1939). From this viewpoint, human beings are the architects of the edifice of mathematics, free to alter its structure or add new building blocks, as long as rules of logic are followed: that is, no blocks suspending in the air. The question of which viewpoint is true has puzzled thinkers for thousands of years, and is still debated among today’s science community. However, despite the apparent difficulty of this question, it is reasonable to argue that the pro-invention (or non-Platonist) view is more worth believing, because the development of mathematics has deep roots in human creativity, and mathematical theories can only imperfectly approximate the physical world.
History reveals that Platonism frequently confined thoughts by setting dogmas on what mathematics ‘is and is not supposed to study’, whereas the non-Platonist belief relieved this burden, increased freedom of mathematicians, and greatly boosted the progress of mathematical research. The expansion of the number concept is a telling example. The ancient Greek sect of Pythagoreans believed so firmly that everything was expressible in rational numbers (i.e. the ratios of whole numbers), that when one of them, Hippasus, discovered that no rational number could multiply itself to give 2, he was severely punished (Taylor, 2014). Today’s math learners, being so familiar with the irrational numbers, would be puzzled by why the Pythagoreans just would not accept square root 2, which can be easily understood as a point on a straight line. Events like this were not unique. The Medieval Europeans typically feared that dealing with zero or Nothingness would violate ‘the rule’ that numbers should measure things that exist. Italian mathematician Fibonacci (1170-1250) was one of the first Europeans to use zero in his works, but he discriminated zero as ‘a sign’ whereas 1-9 were ‘numbers’ (O’Connor and Robertson, 2000). The same fate happened to the imaginary numbers, the square roots of negative numbers. As squares of all magnitudes are positive or zero, imaginary numbers were for hundreds of years regarded as a ridiculous tool used by those who cared to find ‘useless solutions’ of equations (Biggus, 2002). Later, researchers like Euler boldly defined their arithmetic rules and treated them as numbers, and it was soon found that when combined with real numbers (i.e. whole numbers and decimals), they form a sort of two dimensional numbers possessing beautiful patterns and much better properties than real numbers alone. Some complicated calculations would become hugely simplified with them, and they are used in every part of today’s engineering (Kaplan & Kaplan, 2004). The above shows that open-minded pioneers were often held back by the Platonist assumption of a unique and absolute mathematical system. There were too many unnecessary discussions on ‘what numbers are and what they should be’. Non-Platonism, however, views all mathematical concepts as legitimate, the difference being only their ability to yield practical theorems. This philosophy, which prevailed in the 19th Century, has proven to be amazingly fruitful. New mathematical theories, including further expanded number concepts and 4D geometry, emerged at an unprecedented rate. Having seen all these happening, mathematician Richard Dedekind (1888) confidently wrote, in a paper titled precisely What are numbers and what should they be, that ‘Numbers are a free creation of the human mind.’ This paper, discussing how natural numbers could be created with pure logic, later became the foundation of computer science. In fact, many of today’s cutting-edge technologies owe their fundamentals to this movement of mathematics. Nevertheless, it may be argued that playing a more positive role in history does not necessarily mean being true. However, no theory or ideology guarantees its truth, but it is of greater value if it explains things more neatly and surpasses competitors in practical significance. The heliocentric model of the solar system was accepted rather than the geocentric one because it elegantly accounted for telescopic observations and accelerated the development of astronomy. Similarly, non-Platonism explains why mathematics took great steps forward almost every time a rule-breaking concept was introduced, and it pays a mathematician more to believe that the edifice of math is out there for him to freely contribute.
The pro-invention philosophy is not only a source of encouragement for mathematical innovations, but gives a revealing portrait of the role mathematics plays in natural sciences, suggesting that mathematics is not innate, but is only nature’s imperfect approximation. In this view, scientific theories written in mathematical forms describe merely models of nature. Physicists John von Neumann jokingly remarked, ‘With four arbitrary parameters I can produce an equation whose graph fits the look of an elephant’ (Dyson, 2004). Such an equation, of course, is a mathematical model that only superficially resembles an elephant, and has no insight into the underlying natural principles which lead to the formation of an elephant. Unfortunately, almost all scientific theories are more or less ‘fitted elephants’, with special conditions consciously or unconsciously assumed, trivial details ignored, and some arbitrariness imposed to ensure everything works. One of them is Newton’s laws of motion. Though they, expressed in mathematics, look beautiful and perfectly make sense, modern physics has shown that they are only fitted for objects moving slowly, and are seriously wrong at speeds comparable to the speed of light. Another example is the model of friction. Classical physics treats rough surfaces as perfectly smooth first, and then roughness is taken back into consideration in the form of ‘friction coefficients’, an arbitrary index for surfaces’ ability to retard motion. The simple equations derived from this model make such good predictions for motions on surfaces that it is tempting to conclude that they reveal the essence of friction. However, today it is known that friction has a much deeper origin: it results from microscopic interactions of atoms. The friction coefficient, far from being fundamental, is just an emergent property that averages all such interactions. To get a more accurate picture of friction, it should be derived rather than arbitrarily assigned. There are new mathematical equations to do this, but they are again imperfect models, since the behaviours of atoms are not yet completely known. The above discussions should suggest that despite having been applied to sciences so successfully, mathematical descriptions of nature are not necessarily inherent. The reason why they are useful is simply that they were created to be so. In other words, numbers, shapes, and other abstract symbols are manipulated to fit humans’ desire for a mental tool that is both generalised enough to reduce complexity of nature and specialised enough to give accurate predictions (Abbott, 2013). In fact, the idea of mathematics being an approximation encourages scientists to device new mathematical methods to suit their own studies, and even to build diverse mathematical systems that explain the same phenomena. For example, the computation called logarithms were first designed to convert huge data in astronomy into smaller ones (as in log(1,000)=3, log(1,000,000)=6), calculus was developed to tackle motions under varying forces, and in a branch of modern physics called quantum mechanics, the behaviours of matters can be treated by many distinct mathematical approaches, each with its advantages and drawbacks (Rowlands, 2015). In its origin, mathematics could be more of a reflection of how people want to arrange things rather than how things really are arranged.
There are some major arguments against non-Platonism. Some people point out that many pure mathematical theories were initially not created to fit any real being, but were surprisingly found useful in sciences centuries later, so they must represent absolute arrangements in nature. However, as different aspects of nature are related, there is no surprise that a system created for one purpose, when developed by thoughts, can be found useful elsewhere. For example, real numbers for measuring lengths are also used for measuring angles, weights, etc., which can be converted to length by proper apparatus. Another argument, probably thought by Plato but explicitly proposed by G. H. Hardy, is that mathematical statements are either right or wrong independent of human will; therefore 317 is a prime number whether people like it or not (du Sautoy, 2003). But this argument ignores that all mathematical statements are based on intricate systems with carefully chosen axioms and definitions. Prime numbers make sense only if natural numbers and multiplication are defined, which is not trivial, as Dedekind showed in his 1888 paper. If the number concept is an invention, then 317 being a prime number is just part of the invention instead of a universal truth. Obviously, through education, nearly all human minds are equipped with the identical arithmetic system. However, it would be hasty to assume all aliens to have the same system, and therefore the same prime numbers.
In conclusion, mathematics is better seen as an invention than a discovery. The non-Platonist philosophy frees mathematicians, driving them to introduce new ideas. It also convincingly suggests that mathematics only treats models or approximations of physical reality. Today, in light of the requirement for rapid development of mathematics and sciences, the non-Platonist view is arguably more worth believing.
References
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Abbott, D. (2013). The reasonable ineffectiveness of mathematics. Proceedings of the IEEE, 101, 2147-2153.
Biggus, J. (2002). Sketching the History of Complex Numbers. Retrieved 28, Feb 2015 from http://history.hyperjeff.net/hypercomplex.html
Campbell, M. (2013). Mathematics Illuminated. Retrieved 28, Feb 2015 from http://www.learner.org/courses/mathilluminated/about/credits.php
Dedekind, R. (1888). Was sind und was sollen die Zahlen? in Fricke, R., Noether, E., Ore, Ö. (eds.). (1932). Gessamlte Mathematiche Werke. Braunschweig: Vieweg.
du Sautoy, M. (2003). The Music of the Primes. New York: HarperCollins.
Dyson, F. (2004). Turning points: a meeting with Enrico Fermi. Nature, 427, 297.
Hilbert, D. and Bernays, P. (1934). Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften. Berlin, New York: Springer-Verlag.
Kaplan, R. and Kaplan, E. (2004). The Art of the Infinite. London: Penguin.
O’Connor, J. J. and Robertson, E. F. (2000). A history of zero. Retrieved 28, Feb 2015 from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html
Rowlands, P. (2015). How Schoedinger’s Cat Escaped the Box. Singapore: World Scientific.
Taylor, P. (2014). Discovery of irrational numbers. Retrieved 28, Feb 2015 from https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/