Time travel

"Change alone is eternal, perpetual, immortal."

~ Arthur Schopenhauer

 

 

We'd talked yesterday about identity, and how fractured a thing that is. We considered primarily the intersubjective entity - the word, the symbol, the representation of the objective in discourse. Today we're going to look more at the objective entity itself, the Dasein, as we move from a consideration of self-referentiality to infinite recursion.

 

What do we mean by Dasein? The term comes to us from Martin Heidegger, and it literally means "being there." As opposed to simply "being," it denotes being in a particular place - which, for a dynamic entity in a dynamic universe, entails being also at a particular time. A "being," particularly the abstract "being" that we draw upon in discourse, does not have a necessary relation to any other being; the Dasein exists in the context of other entities in time and space. It has coordinates. In an important sense, the Dasein gives meaning to Time and Space - we understand both indirectly by the changes entities undergo through dimensions of time and space. Our consciousness of space is perhaps more direct: our proprioceptive sense tells us how our physical body is oriented in space, and gives us some idea of its relative propinquity to other physical bodies. Our sense of time passing is not as direct, and in fact the naive view of Time as a river flowing from past through present into future can limit our worldview in important ways despite being the most straightforward way to interpret our impressions of the empirical world.

 

Borges, in Funes el memorioso, describes a remarkable character blessed (or cursed) with absolutely perfect recall. This individual's unique worldview creates for him a difficulty with identity - his recall is so perfect that he can recall every single instant of his subjective existence with crystal clarity. He does not need to reference an abstract intersubjective as a placeholder for the vague recollection that must suffice for most of us. He remembers every single instant of perception as its own unique set of entities - the bed or the book or the tree that he saw this morning is, for him, isolated from every other perception of what we would see as "the same" bed or book or tree. Number has no meaning for him; defying arithmetic, he invents his own number system in which each number has its own idiosyncratic name (the number five hundred in his system is known as nine, for example). It may seem that Borges invents Funes merely as a device to investigate the assumptions that underlie our perception of the world; in fact, the neuropsychologist Alexander Luria describes a real-life case with striking similarities, and there are perhaps a dozen such cases in the literature of brain science.

 

There are obviously good reasons to assume that Time does indeed flow in a linear fashion from Past to Future; that our naive impressions are accurate depictions of an empirically real world in which physical entities interact in predictable and measurable ways. Centuries of scientific experiment support this view; but it's worth remembering that the assumptions underpinning science, the axioms of science, predispose us to accept certain sorts of evidence. Inductive reasoning - the scientific habit of extrapolating from known patterns exhibited in the past to predicted patterns expected in the future - suffers from this problem, as David Hume noted: there is no good reason to believe that some relation which has been demonstrated between entities in the past will continue to be demonstrated in the future. One pithy formulation of this is the observation that we can't know that the Sun will rise tomorrow, just because it did today and yesterday and every day before. We can produce all sorts of scientific arguments why it should, but all of those arguments rest on inductive reasoning as well. We must accept axioms on faith, in science as in any system of thought. In fact, an axiom is necessarily not provable within the logic it supports (this isn't Godelian Incompleteness, however; this is a fundamental question of knowability, and one we'll look at later in the company of Fitch and Gettier, among others).

 

If Time is somehow other than linear; if it is, for example, a continuous dimension in which all events we perceive as consecutive are actually simultaneous - if, going one step further, it is a fractal dimension in which all possible events, perceived and unperceived, are simultaneous - then our assumptions about its passing and our motion through it are flawed. The limits of our experience of Time are revealed as precisely that: limits of our experience, and not of Time itself. Paradoxes of infinite recursion encourage us to visualize alternative models of Time that resolve or obviate the paradox - but that we are discussing in another metanow...

Wheels within wheels

“Clouds are not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight line.”
~ Benoit Mandelbrot

 

Of course, there’s more to mathematics than numbers. Mathematics is a way of modeling reality, and there are a range of approaches to that modeling. One of these is to simply interrogate some aspect of reality, and attempt to devise a representation that models the result.

The French mathematician Benoit Mandelbrot took on the formidable challenge of modeling the physical contours of things in the objective realm. He pursued a method of modelling the shapes of clouds and mountains and coastlines, which were not immediately apparent as obeying any coherent mathematical principle. In 1967, he published a paper that asked the innocuous question “How Long is the Coast of Britain?” – it was to prove revolutionary, and would give rise to the concept of fractals. This concept is very important to my own parasimplistic worldview, but of course it has far more important implications than that.

Coastlines, it turns out, are tricky things to measure. If one attempted to measure the coastline in units of, say, 10-meter lengths – approximating the actual contours of the coastline – one would find a shorter result than if one used 1-meter lengths. The 1-meter lengths would give a better approximation, and would be longer as a result. In fact, as Mandelbrot’s paper illustrated, the shorter the unit of measurement becomes, the longer the overall measurement becomes. In the limit of an infinitesimally small unit of measure, the coastline of Britain becomes infinite.

This might seem at first blush to be absurd, but Mandelbrot expanded on this result to show its consistency with a whole family of known mathematical relationships that exhibit the property of self-similarity – that is, the curve viewed at a large scale resembles the same curve at successively smaller scales. With a self-similar curve modeling variable x against variable y, the appearance of the curve between, say, 1 and 2 will be the same as the appearance of the curve between 1.0 and 1.1, or between 1.00 and 1.01, or at any smaller scale of measure. Such curves are said to have a Hausdorff dimension between 1 and 2 – the upper bound, curves of Hausdorff dimension 2, are known as Peano curves and have the property on successive iteration of completely filling the space over which they are measured, after the fashion of a ‘Greek key’ motif. Moreover, Mandelbrot listed several examples of naturally occuring self-similar relations – famously including the leaf fronds of ferns.

Mandelbrot was, like most mathematicians, building on the work of predecessors (including in this case Lewis Fry Richardson, who had tackled the coastline paradox himself and posited a mathematical law governing coastlines that foreshadowed Mandelbrot’s result that coastlines were self-similar). The focus of his 1967 paper, which was to form the basis of his work until his death of pancreatic cancer in 2010, was actually a modern application of a very ancient paradox proposed by the Greek philosopher Zeno of Elea – the dichotomy paradox, famously elaborated in his paradigm of a race between Achilles and the tortoise. Zeno’s paradoxes give us a useful structure for considering parasimplicity and being-in-time (what Heidegger refers to as Sein-in-der-Welt), and we will be returning to them.