Art, Math, and Culture

Kaz Maslanka argues that art expresses culture, whereas math transcends culture. I believe that art and math have much the same relation to culture. Below is a copy of our dialogue.

Embodiment and Metaphor

I recently wrote about Criticisms of Lakoff’s Theory of Metaphor and how some of these criticisms can be addressed by integrating the work of Lakoff and Genter. Perhaps the most controversial aspect of conceptual metaphor theory (CMT) is the idea of embodied cognition, that metaphor is grounded in the sensory experiences of our bodies. I went looking for experimental evaluations of this idea and found some interesting papers.

Reasoning

I was reading the Wikipedia page about Reasoning and the associated Discussion page (by the way, I find the discussion pages are often at least as interesting as the main articles, and sometimes more interesting), and it seems to me that we don’t have a good classification of the various types of reasoning. The page on Logical reasoning describes the three-fold division that is most familiar to me — induction, deduction, and abduction — but the page on Reasoning prefers a two-fold division — induction and deduction. On reflection, neither of these seem adequate to me.

Lakoff and Gentner

In response to my previous post, Criticisms of Lakoff’s Theory of Metaphor, Cosma Shalizi suggested that I should look at some criticisms from Chris and Murphy:

• Explaining the War of the Metaphors, Chris (2007)
• Idioms, Metaphors, and Lakoff, Oh My!, Chris (2006)
• On Metaphoric Representation, Murphy (1996)

Here are my thoughts on these criticisms.

Criticisms of Lakoff’s Theory of Metaphor

Lakoff’s theory of metaphor has been both highly praised and highly criticized. My own thinking about how the mind works has been greatly influenced by Lakoff’s books, yet I also agree with much of what his critics say. I would like to make a case here that his books are worth reading, although much of the criticism is correct.

Meaning, Mapping, Panalogy, and Netflix

“Never swallow anything whole. We live perforce by half-truths and get along fairly well as long as we do not mistake them for whole-truths, but when we do so mistake them, they raise the devil with us.” — Alfred North Whitehead, Dialogues of Alfred North Whitehead

There are dozens of theories about meaning, but they share a common element: meaning (semantics) is about mapping. We understand a thing, we give it meaning, by mapping it to another thing. Furthermore, and this is a crucial point, one mapping is not enough. The more mappings we make, the better we understand. A single mapping only gives us part of the truth.

Symbol Grounding and Proportional Analogy

If symbols must be grounded in perception, how does this grounding happen? How do we learn to create mappings between language and perception? For example, how does the word “rabbit” get tied to the perception (visual, tactile, whatever) of a rabbit? AI algorithms for assigning textual labels to photographs are not yet able to approach human performance on this task. The problem is somewhat similar to statistical machine translation, which exploits parallel corpora to learn mappings between two different languages, although the difference between text and photographs is more extreme than the difference between any two written languages. Perhaps ideas from statistical machine translation are applicable to symbol grounding. The translation algorithm of Lepage and Denoual, based on proportional analogy, seems particularly appropriate, since it makes minimal assumptions about the structures of the languages.

Meditation, Language, and Evolution

There are many kinds of meditation, but a common theme in meditation is control of attention or awareness. In particular, several meditation exercises involve focusing attention on the immediate present, which seems to involve stopping or altering the internal monologue that usually fills our consciousness. It seems to me that this internal monologue, this constant flow of language, is the main thing that distinguishes us from our nearest living relatives, the chimps. Some types of meditation, in stopping the internal monologue, may be altering our consciousness in a way that brings us closer to the consciousness of chimps. (This hypothesis is not intended to denigrate meditation.) I’d like to explore this idea and see where it leads.

Logical Atoms

In predicate logic, the concept red ball is represented as a combination of the concepts of red and ball. We can define the predicate RedBall(x) as (Red(x) & Ball(x)). Logical atomism views the world in terms of compound predicates, such as RedBall(x), that are built up from atomic predicates, such as Red(x) and Ball(x). Good old-fashioned AI (GOFAI) research almost always assumes a kind of logical atomism. Cyc, for example, represents knowledge using a form of logical atomism. Even those researchers who reject GOFAI still tend to assume logical atomism. Statistical and connectionist models of concepts typically view red ball as a combination of red and ball. I believe that we should turn this view on its head. That is, red ball comes first (is more basic, more primitive); red and ball come later (are more complex, more refined).

Math and Art: Differences and Similarities

Mariana Soffer has made a list of some differences between math and art. In a contrarian mood, I will go through the points in this list and discuss the similarities between math and art.

Note: The original source for the following twelve quotations is Kaz Maslanka, Delineations Between Aesthetics of Math and Art. Kaz cites Proceedings of the 2002 Bridges Conference on Mathematical Connections in Art, Music, and Science, page 256. (Note added December 5, 2009.)