No, I’m not talking about the one with a bite out of it on your Mac or the one Granny Smith puts in to her pies. The Banach-Tarski apple states that given a non-hollow object such as an apple, it can be sliced into a finite number of pieces and pieced together to produce two apples of the same volume.
The paradox refutes a function called the Axiom of Choice, which states that given a collection of sets C, one can select one element from each set. The sets inclusive may not have finite parameters, leading this axiom to be one of the most controversial in mathematics.
The Axiom of choice derives on the capaciousness to ascribe existence to a theory or infinity. To assert a “mathematical object exists, is to assert that the number 3 exists”, but all it represents is a placeholder in our imagination, unlike the “solidity of a table or chair”. While the apple in and of itself seems to bely laws of conventional physics such as the Law of the Conservation of Mass or Energy, it does not, as ”only a set of defined volume can assert to have a defined mass” (Schetcher, 2009).
In other words, through human choice, one oft thinks mathematical computations in concrete as supposed to philosophical terms which does not pan out. If one takes a set of numbers to infinity like a line, and slices infinity in half, one will simply just get two sets of smaller infinities.
The concept of Baruch-Tarski’s apple relates to the field of artificial intelligence as one delves into the visualization of infinity, of curves, which in and of themselves are esoteric concepts. On one level, there is one of stereopsis where in which one can distinguish through depth of the lines of the number 3 certainly do not make up a curves of a Frenchman’s moustache. It leads one to question the information encoded in the visual cortex to allow a human to discern the difference.
It turns out that in order to visualize depth, the visual cortex recognizes a face by identifying areas and shapes that are conserved regardless of contrast and lighting. For instance, in a dark night of the moon shining, one can still recognize the same shape of a nose bridge and chin that is conserved in the glaring sunlight of summer. On a study done by Doris Tsao, it turns out that there are about a dozen of facial recognition centers on the temporal lobe of the human brain. It lies in the tempo of life, and in those we recognize and know, is only but a function of space and time, constructed via the data garnished via one’s memories.
One of the most pressing issues with AI is the invariance problem, of how machines are able to delineate a face or any object for the matter, with variations in expression or partial obscurity. Researchers thus implanted electrodes, positing that microstimulation of face patches in the the temporal lobe would cause the distortion of faces and induce the invariance problem; this is proven true as monkeys identified two identical faces as being different when they were the same (Abbott, 2018). On the other hand, microsimulation led to no distortion of non-face objects, with the exception of an apple.
An apple, perhaps in a microstimulated or impaired temporal and visual cortex, is a face.
Researchers have thus utilized rule-based AI to delineate face shapes, and merge it with collections of inputs of photographic data to make facial recognition more accurate. In Tsao’s lab, researcher Le Chang. By taking factors for appearance and factors for face shape, and applied to photographic data, Stephen Le Chang in Tsao’s lab at Caltech showed how human facial recognition itself was a dot product of weighted vector sums. Thus, in the preservation of facial features along the preferred axis along the function of face shape and appearance and its perpendicular vector, all the faces elicited the same response in the cell (Abbott, 2018).
While the current research proves that visualization itself in the course of AI and human beings serves as a linear regression, it fails to take in the other components of the human brain that factor into memory curation. For instance, when one remembers grandma or her apple pies, we remember holistically, the pastoral moments of childhood- the smell of her kitchen, the way she laughed, and the way she played checkers with us when we could barely learn to count.
Memory storages in other regions such as the hippocampus for location, amygdala for fear or stress, olfactory, to muscle and emotive memory are all human contributors to experience that perception AI has yet to fathom. For instance, in the human brain, the he olfactory nerve is only two synapses away from the amygdala which is responsible for certain types of emotions and three synapses away from the hippocampus which is critical for long-term memory (RoedigerIII, 2008). Thus, it is no wonder when in smelling the airs of a passerby of wearing Terre D’ Hermes perfume, that flux of tears of emotions come to our eyes, in remembrance of a lost lover, but for now I’ll settle for grandma’s apple pie. The lack of research in this are and its corresponding counterparts in AI also question how different parts of the brain function together to store predictive and past memory.
For the moment, at least AI can perceive an apple, albeit not the Banach-Tarski sort.
Abbott, A. (2018, December 11). How the brain’s face code might unlock the mysteries of perception. Retrieved from https://www.nature.com/articles/d41586-018-07668-4
RoedigerIII, H. (n.d.). Olfactory Memory. Retrieved from https://www.sciencedirect.com/topics/medicine-and-dentistry/olfactory-memory
Schetcher, E. (n.d.). Axiom of Choice. Retrieved from https://math.vanderbilt.edu/schectex/ccc/choice.html