Are infants young mathematicians?

Logical reasoning paves the way to scientific advances. It is an open question as to how early in life children begin think logically and use inference. The majority of psychologists have reached a consensus that infants are amazingly capable of logical reasoning.

Do infants understand statistics?

Studies have shown that infants can infer sample from population and vice versa (Xu & Garcia, 2008). In the “sample to population” condition, 8-month-old infants were presented with a large opaque box filled with balls. One by one, an experimenter picked a ball from the box and showed it to the infant. Nearly all of the balls that were picked were red, while the remaining balls were white. After this sequence, the front wall of the box was opened and infants were shown that most of the balls inside were either red (expected event) or white (unexpected event). Infants looked longer when most of the balls were white, the unexpected event. In the “population to sample” condition, the front wall of the box was open prior to the sequence and infants were shown that most of the balls inside were red. After that, an experimenter picked a ball from the box. Because infants knew the box contained more red balls. they looked longer when a white ball was picked more frequently from the box (unexpected event) than when a red ball was picked more frequently (expected event). This study demonstrated that infants possess fundamental statistical knowledge and may learn to think rationally from a very young age.

Infants know disjunctive inference.

Here, I describe a second experiment that demonstrates inference in infants (Cesana-Arlotti et al., 2018). Two distinct objects—a flower and a dinosaur— were shown to infants. The top of the flower and dinosaur were identical, and it was only the features of the bottom half that distinguished these two objects. These two objects were then hidden behind an opaque wall, and one of the objects was scooped into a cup. When the cup emerged from behind the wall, only the top portion of the object could be seen protruding from the top edge of the cup. Because the tops of the objects are identical, infants could neither perceive the entire image nor identify which object was scooped into the cup. A dinosaur appeared once the wall was removed. If infants follow the disjunctive rule of inference (which states that "when something is either "A or B" and also "not B", then it must be "A") then they would infer that the object scooped into the cup was a flower. Twelve-month-old infants looked longer when the object in the cup was a dinosaur, because it violated the rule of disjunctive inference. This suggests that even at a young age, infants are already capable of using logic.

These two studies showed that logical reasoning starts from the prelinguistic period of development, and it does not necessarily require formal education.

Can we apply logical reasoning to something that cannot be physically examined?

A woman travelled to Italy and happened to meet a nice guy. They were friendly with one another while she was in Italy. After she returned home, she received a letter from him. He expressed how much he missed her, and that when he shed tears over her, some of them fell into the Adriatic Sea. Japan and the Adriatic Sea are interconnected through several bodies of water. He then said that his tears would arrive close to her someday and he hoped that she would sense his sadness through his tears. She pondered when his tears would arrive at Tokyo Bay. One science journalist set out to answer this challenging question (Kimura, 2003).

To start, the Adriatic Sea, Tokyo Bay, and all the water interconnecting them spans a distance too far to be seen with the naked eye. Thus, we cannot look to nature to answer this question.  Instead, we must use our knowledge of the world to extrapolate a reasonable guess. Instead, one could ask: when would a molecule of water from the Adriatic Sea arrive at Tokyo Bay? H2O has a molecular weight of 18 grams/mole (H=1, O=16, H2O = 2(1) + 1(16)). We can easily calculate the number of water molecules in a tear by the following a few pieces of information taught in high school science class. One mole of H2O contains 6 x 10^23 molecules. A tear, if considered a sphere with a semi-diameter of 0.1 cm, would have a volume of 4/3π x (0.1cm)3 = 4.2 x 10^(-3) cm3. Thus, it would contain 1.4 x 10^20 H2O molecules (6 x 10^23 molecules x 4.2 x 10^(-3)g ÷ 18g). Based on this enormous quantity of molecules in a single tear, it is hopeful that some of them will arrive at Tokyo Bay, despite how vast the planet is.

Based on tide information that has been accumulated over the course of civilization, we are able to calculate that it would take 2,000 years for some molecules of tears released in the Adriatic Sea to arrive at Tokyo Bay, which means that an H2O molecule released into the Adriatic Sea during the Yayoi period is currently in Tokyo Bay.

Logical reasoning leads you to anytime and anywhere

According to the theoretical physicist Steven Hawking, causality, the greatest principle in the universe, is violated by time travel into the past. Thus, while traveling to the future might be feasible, going in the opposite direction can never happen. While we cannot travel back in time, we can view the past through theoretical inference. Theoretical reasoning leads us to any place or time beyond the limitation of causality. It is notable that we are all born with some degree of logical reasoning. Perhaps humans might be scientists by nature.

 Reference

Cesana-Arlotti, N., Martín, A., Téglás, E., Vorobyova, L., Cetnarski, R., & Bonatti, L. L. (2018). Precursors of logical reasoning in preverbal human infants. Science, 359, 1263-1266.

Xu, F., & Garcia, V. (2008). Intuitive statistics by 8-month-old infants. Proceedings of the National Academy of Sciences, 105, 5012-5015.

木村 龍治 (2003). 「自然をつかむ7話」, 岩波書店