The Curve Is the Aerial Line

Our eyes capture a photon by the photoisomerization of retinal. The reaction is shown below.

Hence, there is the digital image on our retina. Since our brain get visual information only from the retina, our first image is the raster image. It must consist of pixels, which is the regular polygon.

The above figure shows three types of tiling the plane with regular polygons: the square, the equilateral triangle, the regular hexagon. Only these three regular polygons can be the pixel of the digital image. That is, curves cannot exist in the digital plane (1). They must be created by our brain. That is, they are fictitious. It is natural to think that they don’t exist. Why do our brain create aerial lines?

The answer is simple from the viewpoint of the evolution. The evolution  has given priority to utility, and it often ignored the logical consistency. Gary Marcus says as follows (2).

Nature is prone to making kluges because it doesn’t “care” whether its products are perfect or elegant. If something works, it spreads. If it doesn’t work, it dies out. Genes lead to successful outcomes tend to propagate: genes that produce creatures that can’t cut it tend to fade away: all else is metaphor. Adequacy, not beauty, is the name of the game.

More specifically, our visual system ignores the logical coherence. Rather, our visual system recognizes patterns, which are important for survival. Probably, the ability of recognizing a circle would be advantageous in the struggle for survival. So, shapes like a circle are categorized as a circle. Similarly, the shape like a human face is recognized as a face. The human being excels in the recognition of the face (2). So, we can easily recognize the face of a specific person (3). However, a face does not have a specific geometrical meaning. Therefore, the evolution created curves without concerning the logical consistency.

1. The Origination of the Revolution of Mathematics

2. Kluge: The Haphazard Evolution of the Human Mind

3. Invariant visual representation by single neurons in the human brain

The Action Potential Is the Original Form of Plato’s One

Plato’s one doesn’t exist in the reality. However, everyone knows that one plus one equals two. If he didn’t know the abstract number, he could not do this calculation. That is, anyone knows Plato’s one by nature. There is a biological entity, which corresponds to Plato’s one. The leading candidate of it is the action potential of the neuron. It is the electrical signal of the neuron. Neurons communicate each other using it.

The most remarkable feature of the action potential is that it is similar to Plato’s one (1). That is, our brain can get only digital information. I quote an impressive paragraph in “From Neuron to Brain” (2).

An important feature of electrical signals is that they are virtually identical in all nerve cells of the body, whether they carry commands for movement, transmit messages about colors, shapes or painful stimuli, or interconnect various portions of the brain. A second important feature of signals is they are so similar in different animals that even a sophisticated investigator (like one of the authors of this book) is unable to tell with certainty whether a photographic record of an action potential is derived from the nerve fiber of whale, mouse, monkey, worm, tarantula, or professor. They are universal coins for the exchange of information in all nervous systems that have been investigated.

Thus, the action potential resembles Plato’s one. This means that Euclidean space is made in the brain because both the input and output signals of the brain are action potentials. Because the brain can get only digital data, it makes the analog image from digital data. The basis of our recognition is digital.

References

1.The Action Potential as the Universal Coin

2. Martin, A. R., Wallace, B. G., Fuchs, P. A., Nicholls, J. G., From Neuron to Brain 4th edition, Sinauer Associates, 2001