Putting two and two together

Updated: Mar 23



The two men credited with the discovery of evolution by natural selection - Charles Darwin and Alfred Russell Wallace arrived at the theory in their own rights. Darwin was left flabbergasted when in 1858, he received a letter from Wallace in which he saw his own theory outlined. Darwin had deferred the publication of his ideas since 1839 (when according to him the theory was clearly conceived).


“ I never saw a more striking coincidence. If Wallace had my M.S. sketch written out in 1842 he could not have made a better short abstract! Even his terms now stand as Heads of my Chapters.”
Darwin’s letter to Charles Lyell, 18 June 1858

Although, Darwin was initially distraught by the prospect of having to share the credit for the discovery of Natural Selection, Darwin and Wallace amicably presented their works jointly at The Linnean Society in July of 1858.


A decade down the line, an ideological rift opened between these congenial minds. Darwin and Wallace found themselves on opposite camps for the first time. The point of contention was the evolution of moral faculties and the human intellect. While Darwin remained faithful to natural selection, Wallace, under the sway of spiritualism resorted to invoking the supernatural to explain intelligence and higher level cognition in humans.


“The higher moral faculties and those of pure intellect and refined emotion are useless to them (savages), are rarely if ever manifested, and have no relation to their wants, desires, or well-being. How, then, was an organ developed so far beyond the needs of its possessor?
Alfred Russell Wallace, Quarterly Review, April 1869

If mathematical thinking is so rare in the animal world then how do we all speak the language of patterns?


Let us take a developmental approach to the problem.


Concept of Space and Time


Much of the mathematical thinking we are capable of is not innate. Although infants have no mathematical concepts, they seem to have the concepts of space and time.


The philosopher Immanuel Kant believed that concepts of space (proximity and distance) and time are pure intuitions. He called them a priori intuitions. They were innate concepts that did not require learning through observation.


There is a certain amount of psychomotor priming that happens throughout early childhood. Our bodies are tuned to work under a certain amount of gravity and atmospheric pressure. Babies also progressively learn the spatial and causal relations between objects.


A child that was raised in outer space may lack dexterity and coordination on earth but, we have no reason to believe that the child would have a flawed concept of space and time.


Children of earth struggling in space.



Set and Constancy of number


The Swiss psychologist Jean Piaget identified some basic concepts that children needed to grasp before they were able to do mathematical calculations. For instance, they needed to understand that two sets are equal in number if and only if there is one-to-one correspondence between their elements.



The set of baby yodas and the set of baby darth vaders are equal in number because they can be neatly paired one-to-one without leaving a remainder. Cute.


Another concept called ‘constancy of number’ is the idea that a set has the same number of elements regardless of their arrangements. The only transformations that change the number is addition or deletion.


“A set or collection is only conceivable if it remains unchanged irrespective of the changes occurring in the relationship between the elements.”
- Jean Piaget, The Child's Conception of Number.

Spatial transformations of quantities (liquids and solids) also follow the principle of conservation. As we shall see, spatial cognition and numerical cognition are intricately connected.


Imagine pouring a liquid from a bowl to a tall and slender tube.


Provided that all of the liquid was poured into the tube, it now has the same quantity of liquid as the bowl before. This ‘notion of invariant quantities’, might seem all too obvious to most most of us. But, children are not born with these concepts. They learn these principles through observation.


With the concepts of set and constancy of number, human children have already vanquished all other species on the planet in math abilities. Our unique ability to gain conceptual knowledge is further augmented by two capabilities.


1. Humans preserve, disseminate and build on existing knowledge aided by language. At any point we have a greatly enhanced wealth of knowledge to draw from.


2. Human babies are uniquely good at learning through the master-apprentice system.


Chimpanzees seem to have traded language and conceptual learning for superior spatial memory.


Numbers in the brain


Many insights into how we do math have come from clinical studies of people, who either lack the ability to do calculations or fail to correctly process numbers. There are many reasons why a person might fail to do arithmetic operations.


An imperfect knowledge of the laws of arithmetic is the most common reason. People who have never formally learned mathematics may have the concept of "more or less" (numerosity). Indeed, even many animals are known to have this concept. But, even to do addition (the easiest kind of arithmetic operation) one needs to follow rules. For instance, one needs to add all numbers in the ones place together but separately from all numbers in the tens place.


A person may have knowledge of all rules required to do arithmetic operations but still fail because of defects in short term memory or conditions like hemianopia (lack of vision in one side of the visual field).


The other set of reasons relate to failure in recognising symbols and numbers. People have reported not being able to recognise symbols after brain injury (asymbolia). For most people numbers appear not as meaningless patterns but as symbols with magnitude information. The number ‘9’ is an inverted ‘6’ but has a different magnitude.


How is ‘6’ different from ‘9’?


Distance Effect


In 1967, Robert Moyer and Thomas Landauer did an experiment where participants were asked to choose the bigger number from a screen with two choices. They discovered that people took lesser time to find the correct answer as the difference between numbers became greater.


Imagine being told to find the longer stick from among two provided. It seems obvious that the task will get easier as the difference in length becomes greater. The same applies to numbers. This phenomenon of increasing ease of comparison between numbers with increasing distance between them is called the distance effect. Therefore, any mental representation of numbers should account for the numerical distance between any two numbers compared.

SNARC Effect

STANISLAS DEHAENE

In 1993, Stanislas Dehaene, Serge Bossini, and Pascal Giraux did a series of experiments where subjects were asked to indicate whether a number presented was odd or even. They found that smaller numbers were responded to faster with the left hand and bigger numbers with the right. The effect, called SNARC (Spatial-Numerical Association of Response Codes) persisted even when subjects performed the task with crossed hands. This suggested a directional relation between space and the mental representation of numbers.


The authors also reported that reading and writing directions seem to influence SNARC effect. They compared the performance of French subjects (reading and writing direction is left to right) to that of Iranian immigrants (Persian is written from right to left). SNARC effect was reduced in Iranian subjects. Cultural biases in directionality have been demonstrated through other interesting experiments since.


In another instructive experiment, Anne Maass and Aurore Russo asked groups of Italian and Arab students to draw a scene to describe a given sentence like “the girl pushes the boy”. Italians who typically write left to write predominantly positioned the girl (subject) to the left of the boy (object). Arab students who typically write right to left predominantly placed the girl to the right of the boy.

Predominant representation of "the girl pushes the boy" in two culturally distinct groups (Maass and Russo, 2003)

Reading habits do not explain all of the variation in mental representation of numbers. There is also a connection between finger counting habits and SNARC effect (Martin Fischer, 2008). Subjects who habitually started counting on their left hands show a normal SNARC effect but subjects who start counting on their right hands do not.


The association between finger sense and mathematical ability is also borne out by clinical studies. Patients suffering from the condition known as Gerstmann Syndrome lose the ability to differentiate their individual fingers and fail to do simple calculations [8]. The discovery of Distance Effect and directional biases like SNARC Effect have led to speculations that we have a Mental Number Line.


The fact that ‘putting two and two together’ is an idiom that means ‘making an obvious logical conclusion’ should be proof of man’s pride in his mathematical aptitude. But, mathematical thinking is in a sense spatial reasoning supercharged by rules that humans have discovered. A sense of numerical distance, direction and an ability to compare (none of them limited to humans) have been taken to fascinating extremes in the way we do math. Just how far can we take it? Although derived from normal visuo-spatial experience, math has given us inklings about higher dimensions and workings of the quantum world. How far will it take us?


REFERENCES

  1. Piaget, J., (1965). The child's conception of number, W. W. Norton & Co.

  2. McCloskey, M., & Caramazza, A. (1985). Cognitive mechanisms in number processing and calculation: Evidence from dyscalculia. Brain and Cognition, 4(2), 171–196.

  3. Cohn, R., (1961), Dyscalculia, Archives of Neurology, 4(3):301–307.

  4. Moyer, R. S., & Landauer, T. K. (1967). Time required for Judgements of Numerical Inequality. Nature, 215(5109), 1519–1520.

  5. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396.

  6. Maass, A., & Russo, A. (2003). Directional Bias in the Mental Representation of Spatial Events. Psychological Science, 14(4), 296–301.

  7. Fischer, Martin. (2008). Finger counting habits modulate spatial-numerical associations. Cortex; a journal devoted to the study of the nervous system and behavior. 44. 386-92.

  8. Gerstmann, J. (1940). Syndrome of finger agnosia, disorientation for right and left, agraphia and acalculia. Archives of Neurology & Psychiatry, 44(2), 398.