I have often said that I get particular pleasure from mathematics that defies common sense expectations. A simple example would be the observation that two things can be the same size even though one of them is contained in the other – like the set of natural numbers and the set of positive even integers. I enjoy these things, not because they suggest that there’s something really strange about mathematics, but rather because they suggest that my common sense is limited and, more importantly, I can get around it.
In a May, 2011 Scientific American blog, John Wilkins addresses the problem of how to reconcile the esoteric nature of science with Darwin’s evolutionary system based on fitness. He reminds us that even Darwin had some difficulty with this, reproducing part of a letter that Darwin wrote in 1881 where he says:
But then with me the horrid doubt always arises whether the convictions of man’s mind, which has been developed from the mind of the lower animals, are of any value or at all trustworthy. Would any one trust in the convictions of a monkey’s mind, if there are any convictions in such a mind?
And I think, why not? I continue to be surprised by what I want to call a prideful misrepresentation of the human world of ideas. The monkey’s mind is looking and giving structure to what it sees in the same way that ours is. It must be the very same perceiving and cognitive mechanisms whose actions are somehow extended and blossom into human culture.
I’ve collected a few references that point to work that may be identifying the springs, creeks, and small rivers through which we build our mental waterways:
Cells that may have helped with continuity:
It was recently reported that a study published in the journal Neuron shows that researchers have identified what they call ‘time cells,’ that is cells in the hippocampus that “robustly represented sequential memories…” Important to the study was the fact that time cell behavior connects related events that are separated by an interruption.
Each cell by itself provided a detailed ‘snapshot’ of the experience, and only at specific moments. But together, the activity from all of the cells filled in the gap,” said coauthor Dr. Christopher MacDonald. The appropriately named “time cells” that were active have much in common with previously described “place cells” that are active when animals are at particular locations in space. The time cells were able to adjust, or “retime,” when the duration of the delay period was altered.
In a summary of their work, the authors say the following:
The hippocampus is critical to remembering the flow of events in distinct experiences and, in doing so, bridges temporal gaps between discontiguous events. Here, we report a robust hippocampal representation of sequence memories, highlighted by “time cells” that encode successive moments during an empty temporal gap between the key events, while also encoding location and ongoing behavior. Furthermore, just as most place cells “remap” when a salient spatial cue is altered, most time cells form qualitatively different representations (“retime”) when the main temporal parameter is altered. Hippocampal neurons also differentially encode the key events and disambiguate different event sequences to compose unique, temporally organized representations of specific experiences. These findings suggest that hippocampal neural ensembles segment temporally organized memories much the same as they represent locations of important events in spatially defined environments.
It has often been discussed (although not so biologically) that our sense of time is a kind of template for the idea of continuity in mathematics, one that largely defines the character of analysis and dynamical systems (which are called flows when seen in real time).
Cells that create space and spatial coordinates:
This work on time cells says that they have much in common with previously discovered cells called place cells. The following is a discussion of place cells from Wikipedia:
Place cells show increased frequency of firing when an animal is in a specific area referred to as the cell’s place field…..When a rat forages randomly in an environment, place fields are only weakly modulated by the direction the rat faces, or not at all. However, when an animal engages in stereotyped behaviour (e.g. shuttling between goal locations), place cells tend to be active in the place field on passes in one direction only.
On initial exposure to a new environment, place fields become established within minutes….In a different environment, however, a cell may have a completely different place field or no place field at all. This phenomenon is referred to as “remapping”. In any particular environment, roughly 40-50% of the hippocampal place cells will be active.
In an environment in which a rat is constrained to walk along a linear track, place fields will often have a directional component in addition to a place component. A place cell that fires at a particular location while the rat walks in one direction along the track will not necessarily fire as the rat visits that location from the other direction.
But there’s more. There are grid cells that provide geometric coordinates for location, generating an internal grid to help with navigation. There are also head direction cells that act like a compass. You can find recent work on grid cells here and a more recent one here. There’s also a youtube video on place cells.
It’s easier to talk about what is often called our accidental number sense (an innate ability to distinguish one from two or three things, or large collections from significantly smaller ones) when we consider intuition in mathematics. And this is probably because, like mathematics, we can see our use of number sense in our conscious mind. But, for me, these cellular level organizations of sensory data are much more provocative and better suited to the idea that we must be using some less than conscious part of us to explore mathematical possibilities.