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A Search for Cognitive Foundations for the Four Classical Lie Groups and Algebras

Andrius Jonas Kulikauskas, Self Learners Network

Physical notions of time and space are based on continuity. Mathematically, these notions are modeled by continuous groups, known as Lie groups, and their local behavior at any point is structured and classified by Lie algebras. In particular, there are four infinite families known as the four classical Lie groups and Lie algebras (An, Bn, Cn, Dn). I share my study of the cognitive foundations for why, intuitively, there are these four classical families and what that tells us about time and space.

My particular interest here is to understand how the mind generates and organizes all of mathematics, its various concepts and branches. Lie groups and Lie algebras bring together analysis and algebra, respectively, by way of distinct interfaces, geometries, spatial intuitions.

The classical Lie groups and Lie algebras are infinite families, and so we can explore how they are able to grow to ever higher dimensions. Lie algebras are studied in terms of their highly symmetric root systems, which are "multiplication tables" for adding vectors. I share my philosophical study of how to make sense of the possiblities for these systems, and especially, how they can make sense in ever higher dimensions. I think of the simplest family, An, as propagating a signal ever forward, yielding a sequence:

1-> 2-> .. n-1-> n-> n+1

But I note that the other families reflect the signal backwards in three possible ways:

1-> 2-> .. n-1-> n-> 0-> -n-> -(n-1)-> -(n-2)

1-> 2-> .. n-1-> n-> -n-> -(n-1)-> -(n-2)

1-> 2-> .. n-1-> n-> -(n-1)-> -(n-2)

I am exploring why, cognitively, there are these three types of mirrors for inverting a signal. This all suggests that we should think of time as inherently a duality of forward and backward directions. I argue that this duality is at the heart of the complex numbers, and that the real numbers and quaternions are less fundamental.

I also investigate how intuitive distinctions between the four classical Lie groups/algebras ground the distinction between four geometries: affine, projective, conformal and symplectic. I relate these geometries to the ways that we can think about a triangle as three paths, as three intersecting lines, as three angles, and as the sweeping out of an oriented area.

I conclude with an application to the study of how our emotions help us distinguish the boundary between our selves and our world, for example, between what is sad and what is surprising. I describe six transformations of the boundary - reflection, shear, rotation, dilation, squeeze, translation - as the possible ways of enriching the four geometries, that is, taking us from a less defined geometry to a more defined one. Here again it is possible to look for how spatial geometry (the boundary of one's self) describes the ways of linking time's forward and backward directions (relevant for emotional expectations).

Andrius Jonas Kulikauskas is an independent researcher whose lifelong quest is to know everything and apply that knowledge usefully. His philosophy is based on documenting, in practice, and deriving, in theory, how cognitive frameworks limit and inform our imagination. His goal is to foster a culture of self learners who interact through the questions they investigate rather than the answers they discover. He has a B.A. in Physics from the University of Chicago and a Ph.D. in Mathematics from UCSD.

Dates: 28 – 29 September, 2018

Conference Venue: Department of Philosophy, Vilnius University, Vilnius, Lithuania

Topic Areas: Open. Philosophy, Arts and Sciences

Space and Time are major topics of research in a vast majority of sciences and philosophy trends. However, at the same time it is difficult to find common points and common language among a variety of different approaches. What is common between notion of space and time in psychology and phenomenology, physics and psychology, analytical and continental approaches? The aim of the conference is to bring together different approaches to space and time issues. The conference will strive towards the cooperation between specialists of different fields that brings together different approaches to the problematic of space and time. We will accept papers dealing with questions of space and time from different perspectives. Interdisciplinary papers are welcome, however, we will also accept papers that deal with topic from the viewpoint of one discipline but are understandable to wider audience.

Abstracts up to 500 words should be sent to: stcvilnius@gmail.com

Abstract should include: title, author(s) affiliation, short CV, email, clearly formulated scientific problem, discussion and conclusion. Abstracts should be written and presented in English