In a previous posting, we examined the status of mathematical objects as creations of the human mind, not objectively existing entities. We also discussed the fact that the science of geometry has expanded from a single system to a great many systems, with no single system being true. So what prevents mathematics from falling into nihilism?
Many people seem to assume that if something is labeled as “imaginary,” it is essentially arbitrary or of no consequence, because it is not real. If something is a “figment of imagination” or “exists only in your mind,” then it is of no value to scientific knowledge. However, two considerations impose limits or restrictions on imagination that prevent descent into nihilism.
The first consideration is that even imaginary objects have properties that are real or unavoidable, once they are proposed. In The Mathematical Experience, mathematics professors Philip J. Davis and Reuben Hersh argue that mathematics is the study of “true facts about imaginary objects.” This may be a difficult concept to grasp (it took me a long time to grasp it), but consider some simple examples:
Imagine a circle in your mind. Got that? Now imagine a circle in which the radius of the circle is greater than the circumference of the circle. If you are imagining correctly, it can’t be done. Whether or not you know that the circumference of a circle is equal to twice the radius times pi, you should know that the circumference of a circle is always going to be larger than the radius.
Now imagine a right triangle. Can you imagine a right triangle with a hypotenuse that is shorter than either of the two other sides? No, whether or not you know the Pythagorean theorem, it’s in the very nature of a right triangle to have a hypotenuse that is longer than either of the two remaining sides. This is what we mean by “true facts about imaginary objects.” Once you specify an imagined object with certain basic properties, other properties follow inevitably from those initial, basic properties.
The second consideration that puts restrictions on the imagination is this: while it may be possible to invent an infinite number of mathematical objects, only a limited number of those objects is going to be of value. What makes a mathematical object of value? In fact, there are multiple criteria for valuing mathematical objects, some of which may conflict with each other.
The most important criterion of mathematical objects according to scientists is the ability to predict real-world phenomena. Does a particular equation or model allow us to predict the motion of stars and planets; or the multiplication of life forms; or the growth of a national economy? This ability to predict is a most powerful attribute of mathematics — without it, it is not likely that scientists would bother using mathematics at all.
Does the ability to predict real-world phenomena demonstrate that at least some mathematical objects, however imaginary, at least correspond to or model reality? Yes — and no. For in most cases it is possible to choose from a number of different mathematical models that are approximately equal in their ability to predict, and we are still compelled to refer to other criteria in choosing which mathematical object to use. In fact, there are often tradeoffs when evaluating various criteria — often, so single mathematical object is best on all criteria.
One of the most important criteria after predictive ability is simplicity. Although it has been demonstrated that Euclidean geometry is not the only type of geometry, it is still widely used because it is the simplest. In general, scientists like to begin with the simplest model first; if that model becomes inadequate in predicting real-world events, they modify the model or choose a new one. There is no point in starting with an unnecessarily complex geometry, and when one’s model gets too complex, the chance of error increases significantly. In fact, simplicity is regarded as an important aspect of mathematical beauty — a mathematical proof that is excessively long and complicated is considered ugly, while a simple proof that provides answers with few steps is beautiful.
Another criterion for choosing one mathematical object over another is scope or comprehensiveness. Does the mathematical object apply only in limited, specific circumstances? Or does it apply broadly to phenomena, tying together multiple events under a single model?
There is also the criterion of fruitfulness. Is the model going to provide many new research findings? Or is it going to be limited to answering one or two questions, providing no basis for additional progress?
Ultimately, it’s impossible to get away from value judgments when evaluating mathematical objects. Correspondence to reality cannot be the only value. Why do we use the Hindu-Arabic numeral system today and not the Roman numeral system? I don’t think it makes sense to say that the Hindu-Arabic system corresponds to reality more accurately than the Roman numeral system. Rather, the Hindu-Arabic numeral system is easier to use for many calculations, and it is more powerful in obtaining useful results. Likewise a base 10 numeral system doesn’t correspond to reality more accurately than a base 2 numeral system — it’s just easier for humans to use a base 10 system. For computers, it is easier to use a base 2 system. A base 60 system, such as the ancient Babylonians used, is more difficult for many calculations than a base 10, but it is more useful in measuring time and angles. Why? Because 60 has so many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) it can express fractions of units more simply, which is why we continue to use a modified version of base 60 for measuring time and angles (and geographic coordinates) to this day.
What about mathematical objects that don’t predict real world events or appear to model anything in reality at all? This is the realm of pure mathematics, and some mathematicians prefer this realm to the realm of applied mathematics. Do we make fun of pure mathematicians for wasting time on purely imaginary objects? No, pure mathematics is still a form of knowledge, and mathematicians still seek beauty in mathematics.
Ultimately, imaginative knowledge is not arbitrary or inconsequential; there are real limits even for the imagination. There may be an infinite number of mathematical systems that can be imagined, but only a limited number will be good. Likewise, there is an infinite variety of musical compositions, paintings, and novels that can be created by the imagination, but only a limited number will be good, and only a very small number will be truly superb. So even the imagination has standards, and these standards apply as much to the sciences as to the arts.
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