A special September issue of Scientific American is organized around questions about what we seem to know, and how or why we may be deceived about the nature of reality. This special September issue has the title: *Truth Lies and Uncertainty.* No doubt the editors are inspired, to some extent, by the challenges to the truth that are happening on a daily basis in our social and political lives. But I was also struck by the close connection between the first three articles in Part 1 of the issue (under the heading Truth) and the questions explored here at *Mathematics Rising*. Part 1 begins with a piece by science writer George Musser, who takes a look at some of the unexpected ways that physicists try to come to terms with the counter-intuitive realities that their theories describe. Among the many interesting conundrums he points to are these:

… according to several mathematical theorems, nothing can be localized in the way that the traditional concept of a particle implies

…Fields, too, are not what they appear to be. Modern quantum theories long ago did away with electric and magnetic fields as concrete structures and replaced them with a hard-to-interpret mathematical abstraction

…The deeper physicists dive into reality, the more reality seems to evaporate.

And, he asks:

…What differentiates physical from mathematical objects or a simulation from the original system? Both involve the same sets of relations, so there seems to be nothing to tell them apart.

One can argue that it is physics’ increasing reliance on mathematics that causes reality to evaporate the way that Musser describes. He does discuss some of the ideas that physicists use to reconcile their mathematics with their reality. One of these is a perspective called Qbism, which is an interpretation of quantum mechanical theory that acknowledges and addresses the role played by the scientist in the development of theory. Also from Musser:

Immanuel Kant argued that the structure of our minds conditions what we perceive. In that tradition, physicist Markus Müller of the Institute for Quantum Optics and Quantum Information in Vienna and cognitive scientist Donald Hoffman of the University of California, Irvine, among others, have argued that we perceive the world as divided into objects situated within space and time, not necessarily because it has this structure but be cause that is the only way we could perceive it. The reality we experience looks the way that it does because of the nature of the perceiving agent.

In the same Part 1 is a piece, written by mathematician Kelsey Houston-Edwards, that addresses the creation vs. discovery arguments about mathematics which is, essentially, the question of whether or not mathematics exists, in some way, independently of human experience. She suggests a useful image:

This all seems to me a bit like improv theater. Mathematicians invent a setting with a handful of characters, or objects, as well as a few rules of interaction, and watch how the plot unfolds. The actors rapidly develop surprising personalities and relationships, entirely independent of the ones mathematicians intended. Regardless of who directs the play, however, the denouement is always the same. Even in a chaotic system, where the endings can vary wildly, the same initial conditions will always lead to the same end point. It is this inevitability that gives the discipline of math such notable cohesion. Hidden in the wings are difficult questions about the fundamental nature of mathematical objects and the acquisition of mathematical knowledge.

I like this image because I find it consistent with what does seem to happen in the research done by mathematicians. But it also suggests a focus for the questions we have about the fundamental nature of mathematical objects, that focus being the significance and nature of the interaction of the thoughts we put forward.

The last piece of this triad is an article by cognitive and computational neuroscientist Anil K.Seth. Seth’s work also proposes that our experience is not really an indication of how things really are, but more what our bodies make of the things that are. His central idea is that perception is a process of active interpretation, that tries to predict the source of signals that originate both outside and within the body.

The central idea of predictive perception is that the brain is attempting to figure out what is out there in the world (or in here, in the body) by continually making and updating best guesses about the causes of its sensory inputs. It forms these best guesses by combining prior expectations or “beliefs” about the world, together with incoming sensory data, in a way that takes into account how reliable the sensory signals are.

For Seth the contents of our perceived worlds are what he calls *controlled hallucinations*, the brains best guesses about the unknowable causes of the sensory signals it receives.

What I find interesting about this discussion of truth is that no one is looking directly at what mathematics is doing, or what mathematics might have to say about the relationship between the brain and the world in which it is embedded through the body. Mathematics has the peculiar character of existing in both the perceiver and the perceived. And maybe this isn’t really peculiar. But there is a reason why mathematics is always crucial to correcting the deceptions present in our experience (as it has done with general relativity and quantum mechanics). In physics mathematics does the heavy lifting of defining the data, giving it meaning, finding the patterns, for example, in what we see of particles, and fields, and their interactions, and everything else. And in cognitive science we now see the mathematical nature of the brain processes that construct our reality. But what I hope to see in mathematics is not just about science. I’m convinced that mathematics can help us see how thought and physical reality are not only related, or interacting, but are somehow the same stuff. I suspect that the physical world is full of thoughts, and ideas are as physical as flowers. But I don’t think we’re clear yet on what physicality really is. Mathematics may be the thing that cracks open the stubborn duality in our experience that is obscuring our view.

Another recent article, unrelated to this truth discussion, added a point to my collection of data about the profoundness of what mathematics seems to tell us about ourselves. This past February, Quanta magazine reprinted an article from Wired.com about possible breakthroughs in understanding how the brain creates our sense of time and memory. The brain processes that create memory have been difficult to identify. For neuroscientists Marc Howard and Karthik Shankar, memory is a display of sensory information in much the same way that a visual image is a display of visual information. But neurons do not directly measure time the way that some neurons measure wavelength or brightness, or even verticality. So Howard and Shankar looked for a way (i.e. equations) to describe how the brain might encode time *indirectly.*

…it’s fairly straightforward to represent a tableau of visual information, like light intensity or brightness, as functions of certain variables, like wavelength, because dedicated receptors in our eyes directly measure those qualities in what we see. The brain has no such receptors for time. “Color or shape perception, that’s much more obvious,” said Masamichi Hayashi, a cognitive neuroscientist at Osaka University in Japan. “But time is such an elusive property.” To encode that, the brain has to do something less direct.

It now looks like the way the brain accomplishes this resembles a fairly familiar strategy in mathematics called the Laplace Transform. The Laplace transform translates difficult equations into less difficult ones by replacing the somewhat complex operation of differentiation with the very familiar operation of multiplication. It’s a mapping that changes time and space relations, described by derivatives, into simpler algebraic relations. Once the algebraic relation is understood, there are mechanisms for translating these solutions back into solutions of the original differential equations.

When Howard heard about Tsao’s results, which were presented at a conference in 2017 and published in

Naturelast August, he was ecstatic: The different rates of decay Tsao had observed in the neural activity were exactly what his theory had predicted should happen in the brain’s intermediate representation of experience. “It looked like a Laplace transform of time,” Howard said — the piece of his and Shankar’s model that had been missing from empirical work.

This model for understanding the neurological components of time, developed by Howard and Shankar, began with just the mathematics. It was a purely theoretical model. But the possibility was demonstrated in the lab work of another neuroscientist, Albert Tsao, working independently of Howard and Shankar. Tsao found, in rats, that firing frequencies for certain neurons increased at the beginning of an event (like releasing a rat into a maze to find food) and diminished over the course of the event. At the start of another trial the firing increased again and diminished again in such a way that each trial could be identified by this pattern of enlivened and diminishing activity of brain activity.

As neuroscientist Max Shapiro sees it:

It’s this coding by parsing episodes that, to me, makes a very neat explanation for the way we see time. We’re processing things that happen in sequences, and what happens in those sequences can determine the subjective estimate for how much time passes.

What I think is important here is that the strategy we developed with the Laplace transform is a strategy the body also employs. This happens all the time, but this seems like a particularly unexpected and intimate instance of it. Mathematics, I expect, is pure structure that exists on the edge of everything that we are and all that there is.

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