###### Evolution

###### Intelligent Design

###### Life Sciences

# Explaining Human Mathematical Ability — Three Evolutionary Hypotheses

*Editor’s note**: Last week we launched the new online college-level curriculum to go with a beloved ID classic, *The Design of Life: Discovering Signs of Intelligence in Biological Systems

*, by mathematician William Dembski and biologist Jonathan Wells. The curriculum is*ID the Future

**free**and**available here**. The 300-page book is hardcover, featuring full-color illustrations and accompanied by a CD with additional materials. Get it on Amazon now. Listen to Jonathan Wells talk about this package of amazing resources on an*podcast. And enjoy the following excerpt by Dr. Dembski and Dr. Wells:*

Humans have many unique cognitive abilities apart from language. Evolutionary theorists have proposed three main types of hypotheses for how these abilities might have evolved: the adaptationist hypothesis, the byproduct hypothesis, and the sexual selection hypothesis. Let’s consider these hypotheses in turn with respect to a specific cognitive ability, namely, mathematics.

## The Adaptationist Hypothesis

How did humans acquire their talent for mathematics? According to the adaptationist hypothesis, mathematical ability conferred a selective advantage on our evolutionary ancestors. Those with better mathematical abilities were as a result better able to survive and reproduce. In other words, they were better able to “adapt” to their environments (hence the term “adaptationist hypothesis”). This hypothesis has a certain plausibility when it comes to the acquisition of rudimentary mathematical abilities like simple arithmetic.

For example, if one of our hunter-gatherer ancestors counted five lions earlier in the day but now sees four of them dead (killed by him and his fellow hunters), a knowledge of basic arithmetic will warn him that one lion is still on the loose. He will thus know to act cautiously, which will translate into a survival and reproductive advantage. But rudimentary mathematical abilities are one thing; developing four-dimensional Riemannian geometries that describe a curved spacetime manifold, as Albert Einstein did, is quite another. It is hardly plausible that abstract mathematics, such as the Einstein Field Equations, confers any immediate survival and reproductive advantage. Moreover, future survival and reproduction is ruled out because evolution does not “look ahead.” So the adaptationist hypothesis breaks down, and other hypotheses are required.

## The Byproduct Hypothesis

According to the byproduct hypothesis, higher cognitive functions like mathematics are not evolutionary adaptations at all. Instead, they are unintended byproducts of traits that are adaptive. Spectacular mathematical abilities are thus said to piggyback on adaptive traits. Pascal Boyer offers such an argument. According to him, some rudimentary ability to count and add is adaptive, but the capacity to do higher-level mathematics is a byproduct of this rudimentary ability. The higher-level capacity is not adaptive by itself; rather, it emerges as a free rider on abilities that are adaptive. But how, exactly, does rudimentary quantitative ability turn into the ability to develop curved spacetime Riemannian geometries or mathematical theories of comparable sophistication? Boyer doesn’t say.^{1}

This is always the weakness of byproduct hypotheses, namely, bridging the gap between what can be explained in standard evolutionary terms (adaptations) and the unexpected “freebies” (byproducts) that come along for the ride. Some free lunches are just too good to be true. And precisely when they are too good to be true, they require explanation. That’s especially true of mathematics: Here we have a human capacity that not only emerges, according to the byproduct hypothesis, from other capacities, but also provides fundamental insights into the structure of the physical universe (mathematics is, after all, the language of physics).^{2} How could a capacity like that arise as the byproduct of a blind evolutionary process, unguided by any intelligence? It is not a sufficient explanation here simply to say that it could have happened that way. Science does not trade in sheer possibilities. If our mathematical ability is the byproduct of other evolved traits, then the connection with those traits needs to be made explicit. To date, it has not been.

## The Sexual Selection Hypothesis

Finally, we turn to the sexual-selection hypothesis. Sexual selection is Darwin’s explanation for how animals acquire traits that have no direct adaptive value. Consider a stag whose antlers are so large that they are more deadweight than defense. Or a peacock whose large colored tail makes it easy prey. How do such structures evolve? According to Darwin, they evolve because they help to attract mates—they are a form of sexual display. Thus, even though these features constitute a disadvantage for survival in the greater environment, the reproductive advantage they provide in attracting mates more than adequately compensates for this disadvantage and provides an evolutionary explanation for the formation of these features.

Geoffrey Miller has applied Darwin’s idea of sexual selection to explain the formation of our higher cognitive functions.^{3} According to him, extravagant cognitive abilities like those exhibited by mathematical geniuses are essentially a form of sexual display. Once a capacity begins to attract mates, it acts like a positive feedback loop, continually reinforcing itself. In the case of cognitive functions, such a positive feedback loop can run unchecked because there are no environmental constraints to impose limits: unlike stag antlers or peacock tails, which can only get so large before their adaptive disadvantage outweighs their ability to attract mates, higher cognitive functions can essentially increase without limits. This, for Miller, is the origin of our higher cognitive functions, and our talent for mathematics in particular.

## The Fundamental Weakness of These Evolutionary Hypotheses

Leaving aside whether mathematical ability really is a form of sexual display (most mathematicians would be surprised to learn as much), there is a fundamental problem with these hypotheses. To be sure, they presuppose that the traits in question evolved, which in itself is problematic. The main problem, however, is that none of them provides a detailed, testable model for assessing its validity. If spectacular mathematical ability is adaptive, as the adaptationist hypothesis claims, how do we determine that? What precise evolutionary steps would be needed to achieve that ability? If it is a byproduct of other abilities, as the byproduct hypothesis claims, of which abilities exactly is it a byproduct and how do these other abilities facilitate it? If it is a form of sexual display, as the sexual selection hypothesis claims, how exactly did the ability become a criterion for mate selection?

In short, the main difficulty with all three hypotheses is that they attempt to account for an existing state of affairs without hard evidence of the factors that brought it about, only speculation. In the case of mathematics in particular, that is an *especially* severe deficit because higher mathematics is not obviously useful when it first emerges. The fact that uses are sometimes found later is, on conventional evolutionary grounds, irrelevant to its emergence. It becomes relevant only if one is justified in thinking that there is purpose in nature.

## Intelligent Design?

Certainly, if evolution is true, then one of these hypotheses or some combination of them is likely to account for our ability to do mathematics. But even if evolution *is* true, in the absence of a detailed, testable model of how various higher-level cognitive functions emerged, these hypotheses are scientifically sterile. On the other hand, from an intelligent design perspective, mathematics is readily viewed as an inherent feature of intelligence and rationality. Moreover, the fact that the mathematical theorems we prove mirror the deep structure of physical reality suggests that intelligence is fundamental to nature and not merely an accidental or historical byproduct of blind material forces. The intelligence underlying nature as reflected in mathematics is a theme explored by Eugene Wigner, who referred to the “unreasonable effectiveness” of mathematics in elucidating nature.^{4}

## Number Sense in Animals

Many animals have a basic ability to know the difference between more and less, or many and few. Rhesus monkeys and chimpanzees appear to pay more attention to a quantity if it has changed than if it hasn’t. According to M. D. Hauser, captive rhesus monkeys have been taught to understand ordinal relations from 1 to 9, but only after hundreds of training trials in conditions that are not duplicated in the wild.^{5} Essentially, after six months of training, some rhesus monkeys were accurate 50 percent of the time in identifying an ascending or descending order from 1 to 9.^{6} A weakness of this research is the high level of human interference, a point often overlooked in evolutionary literature (though not by Hauser). The monkeys develop this skill under intensive training by humans. It is unlikely that they would do so otherwise, because almost any non-destructive use of the average wild monkey’s time would be better and more immediately rewarded in nature. This fact tells *against* an adaptationist hypothesis in explaining even the most basic arithmetic skills, never mind abstract mathematical skills that typically only find a use after they have emerged apart from any survival goal.

**Notes**:

(1) Boyer makes this argument in *Religion Explained: The Evolutionary Origins of Religious Thought *(New York: Basic Books, 2001). In attempting to account for higher cognitive functions, Boyer is concerned not just with mathematics but also with art, religion, and ethics. For another byproduct approach to higher cognitive functions, see Steven Mithen, *The Prehistory of the Mind: The Cognitive Origins of Art, Religion, and Science* (London: Thames & Hudson, 1996). Mithen sees higher-level functions like mathematics as the byproducts of a “cognitive fluidity” that is adaptive in the sense that it facilitates the coordination and communication of various lower-level cognitive modules.

(2) See especially Mark Steiner, *The Applicability of Mathematics as a Philosophical Problem* (Cambridge, Mass.: Harvard University Press, 1999).

(3) See his book *The Mating Mind: How Sexual Choice Shaped the Evolution of Human Nature* (New York: Doubleday, 2000).

(4) See Eugene P. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” *Communications in Applied Mathematics* 13 (1960): 1. For a deeper exploration of this theme, see Steiner, *The Applicability of Mathematics as a Philosophical Problem*.

(5) M. D. Hauser, “What Do Animals Think about Numbers?” *American Scientist* 88 (2) (2000): 144–51.

(6) Beth Azar, “Monkeying Around with Numbers,” *Monitor on Psychology: Science Watch* 31(1) (January 2000): available online here (last accessed June 7, 2006).

*Photo credit: © goodmanphoto — stock.adobe.com.*