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.

The Boston Globe recently had a story on this topic:

Thinking literally: The surprising ways that metaphors shape your world
Drake Bennett

Drawing on philosophy and linguistics, cognitive scientists have begun to see the basic metaphors that we use all the time not just as turns of phrase, but as keys to the structure of thought. By taking these everyday metaphors as literally as possible, psychologists are upending traditional ideas of how we learn, reason, and make sense of the world around us.

Several researchers are mentioned, but no references are given, so I tracked down some of the references:

Experiencing Physical Warmth Promotes Interpersonal Warmth
Lawrence E. Williams and John A. Bargh

… we hypothesized that experiences of physical warmth (or coldness) would increase feelings of interpersonal warmth (or coldness), without the person’s awareness of this influence …

Contemporary cognitive linguists have advanced similar arguments that people conceptualize their internal, mental worlds by analogy to the physical world (9–13). Applied to the question of how warm objects can produce the same affective states as a “warm” person, embodiment theorists have noted how objects and events that produce the same quality of affective response are associated (categorized) together in memory (14).

In summary, experiences of physical temperature per se affect one’s impressions of and prosocial behavior toward other people, without one’s awareness of such influences.

Cold and Lonely: Does Social Exclusion Literally Feel Cold?
Chen-Bo Zhong and Geoffrey J. Leonardelli

Metaphors such as icy stare and cold reception depict social exclusion using cold-related concepts; they are not to be taken literally and certainly do not imply reduced temperature. Two experiments, however, revealed that social exclusion literally feels cold.

The Smell of Virtue: Clean Scents Promote Reciprocity and Charity
Katie Liljenquist, Chen-Bo Zhong, and Adam D. Galinsky

Two experiments demonstrated that clean scents not only motivate clean behavior, but also promote virtuous behavior by increasing the tendency to reciprocate trust and to offer charitable help. Capitalizing on the fact that abstract concepts are often symbolically derived from the concrete environment (Emerson, 1836), our results suggest that olfactory cues can trigger virtuous behaviors that are related to cleanliness at only a symbolic level. The link from cleanliness to virtuous behavior appears to be a nonconscious one: in neither experiment did participants recognize an influence of scent on their behavior, and in Experiment 2, perceived cleanliness did not differ by condition nor correlate with the effects.

Weight as an Embodiment of Importance
Nils B. Jostmann, Daniël Lakens, Thomas W. Schubert

Four studies show that the abstract concept of importance is grounded in bodily experiences of weight. Participants provided judgments of importance while they held either a heavy or a light clipboard. Holding a heavy clipboard increased judgments of monetary value (Study 1), and made participants consider fair decision-making procedures to be more important (Study 2). It also caused more elaborate thinking as indicated by higher consistency between related judgments (Study 3) and by greater polarization between strong versus weak arguments (Study 4). In line with an embodied perspective on cognition, these findings suggest that, similar to how weight makes people invest more physical effort in dealing with concrete objects, weight also makes people invest more cognitive effort in dealing with abstract issues.

Here are the home pages of some people who are doing research in this area:

• John A. Bargh
• Daniel Casasanto
• Lawrence Williams
• Chen-Bo Zhong
• Nils B. Jostmann

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.

Wikipedia and Stanford Encyclopedia of Philosophy have articles on the following types of reasoning:

• Deductive reasoning
• Inductive reasoning
• Abductive reasoning
• Statistical reasoning
• Defeasible reasoning
• Bayesian reasoning
• Analogical reasoning
• Dialectical reasoning
• Common sense reasoning
• Retroductive reasoning
• Practical reasoning
• Informal reasoning
• Moral reasoning

The two-fold division of deductive versus inductive really amounts to deductive versus non-deductive, which isn’t fair to the various kinds of non-deductive reasoning.

I guess we should start by defining reasoning. I don’t like the definition on the Reasoning page:

Reasoning is the cognitive process of looking for reasons for beliefs, conclusions, actions or feelings.

I prefer the following definition, which is a variation of a definition that once appeared on the Inference page:

Reasoning is the process of deriving consequences from premises.

The various types of reasoning mentioned above can all be viewed as algorithms, or broad classes of algorithms, that take premises as input and generate consequences as output.

Deductive reasoning seems to have a special status in this list of types of reasoning, perhaps because deductive reasoning is the foundation of mathematics. There are various logical systems for deductive reasoning, but they all share this defining property: If the premises are true (and consistent, although there is paraconsistent logic) and we follow the rules of the given deductive system (e.g., first-order predicate calculus), then the consequences will also be true. This is a very attractive (seductive, even) property for a system of reasoning. (Although deductive reasoning is vital to math, the problem of interestingness in math is arguably more interesting than the problem of truth.)

Statistical reasoning also has a kind of guarantee: If we know the probabilities of the premises (and these probabilities are consistent) and we follow the rules of the given statistical system, then we can calculate the probabilities of the consequences.

It’s tempting to classify the various types of reasoning according to the kinds of guarantees that we have about them, but I think this is the wrong path to take. In deductive reasoning, how do we know that the premises are true? In statistical reasoning, how do we know what probabilities to assign to the premises? These questions lead to endless debate, but the ultimate answer is that we don’t know. This limits the value of these guarantees.

I prefer to take an evolutionary perspective on knowledge. We don’t know anything for certain; everything is open to doubt, but incremental doubt: doubt one thing at a time. We can even doubt the (standard, traditional) rules of deductive reasoning, such as the law of the excluded middle.

From an evolutionary perspective, the various kinds of reasoning listed above are like species of animals that have evolved over time. It might be fruitful to look at the evolutionary tree for these species of reasoning, just as cladistics is useful for biology. From this perspective, it is interesting that both deductive reasoning and analogical reasoning can be traced back to Aristotle.

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.

This is what I make of Murphy’s paper: What Murphy calls Structural Similarity (Sections 3.3 and 6) is the right approach to metaphor. Murphy’s idea of Structural Similarity is that (something like) Gentner’s Structure Mapping Theory (SMT) [1] is the underlying mechanism for Lakoff’s Conceptual Metaphor. I support this view [2], as does Gentner [3,4,5]. The other views Murphy talks about, strong and weak metaphoric representation (Sections 3.1 and 3.2), are strawmen.

However, it seems to me that Murphy does not fully understand Gentner’s work. In Section 3.3, he says, “The structural similarity view is not a theory of novel verbal metaphor.” This is wrong. The structural similarity view is exactly a theory of novel verbal metaphor [3]. Murphy doesn’t seem to understand that the point of Gentner’s SMT is to transfer knowledge from one domain to another. Murphy seems to believe that the two domains (source and target) must be completely structured before SMT can be applied, and that SMT does not add any new structure. If this were so, then the mapping process in SMT would be no more than an entertaining exercise, rather than a method to generate new knowledge.

Summary: the bulk of Murphy’s paper is spent on presenting and refuting strawmen. His proposed solution is the right way to go, but he doesn’t fully understand his own proposal.

[1] Structure-mapping: A theoretical framework for analogy, Gentner (1983)
[2] The latent relation mapping engine, Turney (2008)
[3] The career of metaphor, Bowdle and Gentner (2005)
[4] Metaphor is like analogy, Gentner, Bowdle, Wolff, and Boronat (2001)
[5] Metaphor as structure-mapping, Gentner and Bowdle (2008)

In Explaining the War of the Metaphors, Chris writes:

“I and others think that CMT and blending would benefit greatly from research on analogy. If these theories had a testable model of how mappings between conceptual domains are formed, interpreted, and used — a model that structure mapping theory can provide — it would be possible to make some concrete, experimentally testable predictions about things like inference, object recognition, etc.”

I completely agree that CMT (conceptual metaphor theory) would benefit greatly from research on analogy; in particular, the research of Gentner and her colleagues.

In Idioms, Metaphors, and Lakoff, Oh My!, Chris writes:

“Sometimes I forget that not everyone who happens upon this blog today has been reading it from day one (I mean come on, why haven’t you?). It surprises me, then, when people tell me they’ve seen no evidence that George Lakoff and Mark Johnson’s conceptual metaphor theory is, well, wrong.”

I disagree that Lakoff and Johnson’s theory is (wholly) wrong. Some details may be wrong, but I think that many of the core ideas are correct and furthermore consistent with Gentner’s theories and experiments.

It seems to me that Chris is arguing two incompatible claims: (1) Gentner’s theory is the proper foundation for Lakoff’s theories. (2) Lakoff is wrong. (If not incompatible, these claims are at least dissonant.)

In Metaphor as structure-mapping, Gentner and Bowdle write:

“We have suggested that metaphor is like analogy — that the basic processes of analogy are at work in metaphor. Specifically, we suggest that structural alignment, inference projection, progressive abstraction, and re-representation are employed in the processing of metaphor and simile. This view can help resolve some tensions in the field: for example, on this view, metaphor both reflects parallels (Murphy, 1996) and creates new similarities (Lakoff, 1990) between the domain compared, via structural alignment and candidate inferences, respectively.”

I agree with Gentner that her view “can help resolve some tensions in the field”. It puzzles me that Chris simultaneously agrees with Gentner and yet seems to actively work to increase tensions in the field. Instead of attacking Lakoff, why doesn’t Chris work to integrate Lakoff’s and Gentner’s theories, as I attempt to do?

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.

These are the books by Lakoff and various co-authors that I have read:

• Metaphors We Live By, Lakoff and Johnson
• Women, Fire, and Dangerous Things, Lakoff
• Where Mathematics Comes From, Lakoff and Nuñez
• Philosophy in the Flesh, Lakoff and Johnson
• More than Cool Reason, Lakoff and Turner

I enthusiastically recommend all five of these books to anybody who has a strong interest in either cognition or language.

Below are excerpts from four opinions of the work of Lakoff and co-authors. After each excerpt, I give my reaction to the opinion.

I dislike Lakoff’s theory of metaphor, especially when compared to Sperber and Wilson’s. His work is deeply, purely speculative, which is fine, but he seems not to care at all about experimental controls, or even admit they’re an issue, which is not. And his manner of writing is most unpleasant, principally because it’s so grandiose (he’s overthrowing the whole tradition of western thought) and so dogmatic (e.g., he often writes “cognitive science has shown that” when what he means is “as I have often claimed, and many others vehemently denied”). I half suspect, given the subjects Lakoff is writing on, that he’s deliberately positioning himself to be the 21st-century version of Freud, the man who provides a educated non-specialists with a scientific-sounding vocabulary for mental life. But I hasten to add that (a) I really have no evidence for that, and (b) Lakoff is in person polite, affable and well-spoken. Still, it’s very hard for me to force myself all the way through one of his books. — Cosma Rohilla Shalizi, Analogy and Metaphor

It is true that Lakoff does not seem interested in experimental controls, but other researchers have been working to address this problem. See Chapter 6 of Philosophy in the Flesh for many references. It is also true that Lakoff is often grandiose and dogmatic, and he neglects to cite much related work, such as the work of Dedre Gentner and her colleagues. However, his basic claims, stripped of their abrasive presentation, are insightful and fruitful. If you can focus on the ideas and ignore the style of presentation, there is much to learn from Lakoff’s books.

No one has a problem with the idea that the lens of an eye and the lens of a telescope are two instances of the general category “lens,” rather than the telescope being a “metaphor” for the eye. Nor is there anything metaphorical going on when we refer to “the genetic code”: a code by now is an information-theoretic term for a mapping scheme, and it subsumes cryptograms and DNA as special cases. But do cognitive psychologists use the computer as a “metaphor” for the mind, or (as I believe) can it be said that the mind literally engages in computation, and that the human mind and commercial digital computers are two exemplars of the category “computational system”?

So the ubiquity of metaphor in language does not mean that all thought is grounded in bodily experience, nor that all ideas are merely rival frames rather than verifiable propositions. Conceptual metaphors can be learned and used only if they are analyzed into more abstract elements like “cause,” “goal,”, and “change,” which make up the real currency of thought. And the methodical use of metaphor in science shows that metaphor is a way of adapting language to reality, not the other way around, and that it can capture genuine laws in the world, not just project comfortable images onto it.

[...]

Though metaphors are omnipresent in language, many of them are effectively dead in the minds of today’s speakers, and the living ones could never be learned, understood, or used as a reasoning tool unless they were built out of more abstract concepts that capture the similarities and differences between the symbol and the symbolized. For this reason, conceptual metaphors do not render truth and objectivity obsolete, nor do they reduce philosophical, legal, and political discourse to a beauty contest between rival frames.

Still, I think that metaphor really is a key to explaining thought and language. The human mind comes equipped with an ability to penetrate the cladding of sensory appearance and discern the abstract construction underneath — not always on demand, and not infallibly, but often enough and insightfully enough to shape the human condition. Our powers of analogy allow us to apply ancient neural structures to newfound subject matter, to discover hidden laws and systems in nature, and not least, to amplify the expressive power of language itself. — Steven Pinker, The Stuff of Thought

Pinker agrees that metaphor is important, but he believes that abstraction is more important. He claims that abstraction captures the similarities and differences that are the basis of metaphor. However, Pinker neglects to explain how abstraction works. I believe that metaphor (more precisely, analogy-making) is the basis for abstraction. First we observe an analogy, such as the analogy between water waves and sound waves, and then we form an abstraction, wave theory, which includes both types of waves. Later we add light waves. Abstraction does not subsume metaphor, because analogy precedes abstraction and is the basis for abstraction.

Whereas the authors devote 44 pages to Chomsky, they cover all of “Anglo-American analytic philosophy” in 29 pages, while lumping together Frege, Russell, Carnap and the Vienna Circle, Quine, Goodman, Davidson, Putnam, Kripke, Montague, and Lewis. In the same chapter, they continue with ordinary language philosophy (Strawson, Austin, and the later Wittgenstein), which they consider to be based on the same metaphors. Yet these philosophers have expressed widely divergent views on the embodiment of mind, the nature of language, and Chomsky’s theory of autonomous syntax. By drawing finer distinctions, the authors might have claimed some of them as potential allies against Chomsky’s position. — John Sowa, Review of Philosophy in the Flesh

Like Shalizi, Sowa criticizes Lakoff’s presentation. Yes, Lakoff distorts the work of others and exaggerates the novelty of his own work. Yes, his ideas could be presented more clearly, more fairly, and with better discussion of related work. Nonetheless, if you can ignore these problems with presentation, you will find many interesting ideas in the work of Lakoff and his co-authors.

Lakoff and Nuñez have a very interesting general framework for approaching their topic: mathematics arises by the extension of innate human capacities (e.g. subitization) or basic universals of human experience (spatial and motor experience), and the means of extension is cognitive metaphors which preserve the basic inferential structure of the source domain. The first few chapters provide a plausible sounding, perhaps workable account of arithmetic, simple logic and set theory, but one that they should have developed in far more detail (e.g. their account of intersection in terms of container schemas is criminally underexplained).

[...]

In short, they have very interesting ideas, but the mass of technical vagueness and blunders, plus the big strawman that is their “philosophical” argument, suggests that they are more interested in passing off as intellectual revolutionaries among the pop-science book audience than in contributing to our understanding of the topic. — Idiosyncrat, Review of Where Mathematics Comes From

The final sentence of this review serves as a good summary: the ideas are very interesting, but the presentation of the ideas is impaired by a desire to seem revolutionary, at the cost of fairness, balance, rigour, and careful discussion of related work. For me, ultimately, ideas trump presentation. Until there is a better source for these ideas, I recommend Lakoff’s books.

“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.

We understand things (words, events, perceptions, people, signs) by relating (connecting, mapping) them to other things. This analogy-making is how we understand both high-level ideas and concepts and low-level perceptions:

“We have repeatedly seen how analogies and mappings give rise to secondary meanings that ride on the backs of primary meanings. We have seen that even primary meanings depend on unspoken mappings, and so in the end, we have seen that all meaning is mapping-mediated, which is to say, all meaning comes from analogies.” [emphasis added] — Douglas Hofstadter, I am a Strange Loop

But one analogy is not enough. No analogy is perfect, and we compensate for their imperfections by using multiple analogies, and by blending analogies together. Marvin Minsky calls this panalogy (parallel analogy):

“If you ‘understand’ something in only one way, then you scarcely understand it at all—because when you get stuck, you’ll have nowhere to go. But if you represent something in several ways, then when you get frustrated enough, you can switch among different points of view, until you find one that works for you!” — Marvin Minsky, The Emotion Machine

If you want to learn more about how we blend analogies, I highly recommend The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities, by Gilles Fauconnier and Mark Turner.

Perhaps the most important lesson from the Netflix Prize has been that many models are better than one:

• “the biggest lesson for ML from the Netflix contest has been the formidable performance edge of ensemble methods” — John Langford, Netflix nearly done
• “We have learned that ensemble methods are the solution for more accuracy.” — Daniel Lemire, After Netflix? What next?

I view this as confirmation of the Panalogy Principle: we don’t understand anything until we understand it many ways. There are no whole-truths, but we can get by reasonably well with a large number of half-truths.

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.

There is a view that the meaning of words must be grounded in perception:

• Symbol grounding and meaning: A comparison of high-dimensional and embodied theories of meaning
• Perceptual Symbol Systems

But how do we connect a word with a perception? Quine gives the following example: suppose we hear a person say “gavagai” in the presence of a rabbit. How do we know that “gavagai” means “rabbit”? Other possible interpretations of “gavagai” are “Lo, food”, “Let’s go hunting”, and “There will be a storm tonight”. The possibilities are endless.

It is interesting that Quine links the symbol grounding problem to the problem of translation. In Quine’s thought experiment, a linguist must translate “gavagai” into English. Perhaps recent progress with machine translation algorithms is applicable to the symbol grounding problem?

Lepage and Denoual use proportional analogies to derive translations from parallel corpora. A proportional analogy has the form A:B::C:D, meaning “A is to B as C is to D“. For example, quart:volume::mile:distance means “quart is to volume as mile is to distance”. Consider the following proportional analogy:

A = “This is a rabbit.”

B = “This is a fox.”

C =

D =

The analogy A:B::C:D helps us to map “rabbit” to the image of the rabbit, whereas A and C alone leave the mapping indeterminate. Comparing A and B, we see the shared structure “This is a X.” and we note that X = “rabbit” for A and X = “fox” for B. Likewise, comparing C and D, we see the shared backgrounds and note the differing foregrounds. This helps us map the foreground of C to “rabbit” and the foreground of D to “fox”. Given only A and C, we have no reason to pick out “rabbit” from the sentence “This is a rabbit” in A and we have no reason to pick out the foreground rabbit from the background grass in C.

The core idea here is that we do not ground symbols in perceptions by noting correlations between symbols and perceptions; rather, we note meta correlations between relations between symbols (e.g., the relation between A and B) and relations between perceptions (e.g., the relation between C and D).

The ability to consider differences between differences is important because it lies at the heart of our abilities to solve new problems. This is because these “second-order-differences” are what we use to remind ourselves of other problems we already know how to solve. Sometimes this is called “reasoning by analogy” and is considered to be an exotic or unusual way to solve problems. But in my view, it’s our most ordinary way of doing things.
— Marvin Minsky, The Society of Mind, 1988

Related work: Deb Roy.

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.

We might think of the human mind metaphorically as a chimp mind with language processing bolted on top. A thin layer of language processing runs over a substrate of nonlinguistic thinking. The language layer has a big impact on our consciousness, but a great deal of thinking takes place in the chimp substrate, although it is less accessible to introspection.

Meditation exercises seem to alter the flow of this layer of linguistic processing. Often the alteration makes us feel better: happier, more relaxed, calm, alert, more aware of other people and things around us. It seems that the language layer has benefits and costs. The benefits include improved logic, analysis, planning, efficiency, goal directedness, and rational communication. The costs include psychological stress, worry, obsession, fear, distraction from the immediate present, and reduced emotional, nonverbal communication.

I’m glad that I am a human, not a chimp, but I can see some advantages to shutting down or moderating the language layer. This suggests to me that we humans are in the middle of an ongoing evolutionary process; that language has not yet been fully integrated with our chimp cores. It seems to me that there is no fundamental conflict between the chimp substrate and the language layer; in principle, it should be possible for these layers to work together smoothly, without stress and conflict. Just as the evolution of the female pelvis has not caught up with the evolution of baby head size, so the evolution of the human brain has not fully integrated the language layer.

Meditation might be viewed as a step towards better integration of the chimp layer with the language layer. For example, knowing how to shut down the language layer when it is not needed could be a valuable skill. For some tasks, such as visual memory tasks, it seems that language gets in the way. Perhaps savant syndrome illustrates what might be possible if we can circumvent the language layer and directly access lower layers.

If this hypothesis is correct, I wonder whether it suggests any new insights into meditative practices. I’ve done some searching online for related ideas. Perhaps the closest match is Kevin Langdon’s analytical tracking, although it is difficult to determine from the short descriptions I have found. This is also related to the pop psychology left-brain versus right-brain distinction. What I’m wondering is whether it might be useful to treat meditation as a tool for chimp-layer-language-layer integration, rather than (or in addition to) the more customary use of meditation for (other) spiritual purposes.

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).

In Latent Semantic Analysis (LSA), we represent the semantics of red and ball with vectors, in which the elements are derived from the frequencies of the terms red and ball in various contexts. For those who are familiar with logic, it is natural to think of representing red ball by some mathematical operation on the vectors for red and ball. For example, we might add the red vector to the ball vector. One problem with this idea is that addition is not sensitive to order; thus house boat and boat house would have the same vector, although they have different meanings. One solution to this problem is a mathematical operation on vectors that is sensitive to order, such as the tensor product or the circular convolution.

Several papers explore this vector combination approach to representing compound predicates:

• Plate (1991), Holographic reduced representations: Convolution algebra for compositional distributed representations
• Plate (1994), Distributed representations and nested compositional structure
• Smolensky (1994), Grammar-based connectionist approaches to language
• Plate (1995), Estimating analogical similarity by vector dot-products of holographic reduced representations
• Wilson, Street, and Halford (1995), Solving proportional analogy problems using tensor product networks with random representations
• Jones and Mewhort (2007), Representing word meaning and order information in a composite holographic lexicon
• Widdows (2008), Semantic vector products: Some initial investigations

Plate (1995) describes his approach as follows:

Suppose we have distributed representations for the concept “circle”, “triangle”, “small”, and “large”. We can represent a small circle by superposing the patterns for “small” and “circle”. However, when we try to represent a small circle and large triangle we have the problem that the superposition of the four patterns is ambiguous – the information that small is associated with triangle and large associated with circle is lost – it could be a large circle and a small triangle. The same problem arises when we try to represent conceptual relations. Suppose we have a predicate representation for “Spot bit Jane”: bite(spot, jane). Spot is the agent of this relation, and Jane is the object (or patient). A distributed representation of this relation must be careful to preserve the information about which person is associated with which role (agent or object) so that there is no confusion with “Jane bit Spot”. I refer to associations between roles and fillers as role/filler bindings.

The assumption here is that we build up the complex structure bite(spot, jane) from simple atomic elements bite, spot, and jane. But I suggest that the atomic element is bite(spot, jane), and that we construct bite, spot, and jane from this atomic element.

Suppose we have a third-order tensor of the form pattern × word × word. The predicate P(x,y) is a pattern (“x bit y“), x is a word (“spot”), y is another word (“jane”), and the triple <P(x,y),x,y> is a cell (<”x bit y“,”spot”,”jane”>) in this third-order tensor. The concept “bite” corresponds to the slice (a matrix cut out of the tensor) <”x bit y“,*,*>, the concept “spot” is the slice <*,”spot”,*>, and the concept “jane” is the slice <*,*,”jane”>.

In this representation, the atomic elements (tensor cells; scalars) are whole events (“I see a red ball”, “Spot bit Jane”), and abstract concepts (“red”, “bite”) are complicated structures (tensor slices; matrices), composed of these atomic elements. This turns the usual picture upside-down. There is nothing “simple” or “atomic” about the concept “circle”. It is a complex structure, composed of all of our atomic experiences of events that contained some aspect of circularity.

Using a vector combination approach, Jones and Mewhort (2007) report a score of 57.81% (see page 18) on multiple-choice synonym questions from the TOEFL (Test of English as a Foreign Language). Using the opposite approach (Turney, 2007), a third-order pattern × word × word tensor achieves a score of 83.75% (see page 22).

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.

Difference #1: Mathematical truths are discovered. Artistic truths are mediated.

The nature of truth in math is a difficult philosophical problem. Truth in art is perhaps even more problematic. But one lesson we have learned from Doug Lenat’s AM (Automated Mathematician) is that interestingness is arguably more important than truth. It is easy to write a program that generates an endless stream of mathematical truths (1+1 = 2, 1+2 = 3, 1+3 = 4, …); it is much harder to write a program that generates an endless stream of interesting mathematical truths. In this respect, art is much like math: It is much harder to make interesting art than to make true art. In both art and math, truth is (arguably) required for interestingness, but interestingness is more interesting than truth. (Computers can generate art, but is it interesting art?)

It might be said that math is discovered, whereas art is created, but discovery and creation are both aspects of evolution. Mathematical knowledge evolves. Artistic techniques and methods evolve. In both cases, differential fitness is determined by the degree of interestingness.

Difference #2: Mathematicians generally agree on what is mathematically correct. Artists generally have no idea what is artistically correct.

The first difference concerns the origins of math and art (where does truth come from?). The second difference concerns validating math and art, after the act of discovery or creation is complete (is it really true?).  There is more consensus about truth in math than about truth in art, but, again, truth is relatively trivial, in contrast with interestingness. Arguably, the level of agreement among mathematicians about what is interesting in math is similar to the level of agreement among artists about what is interesting in art.

Difference #3: Math illuminates the supportive skeletal structure of thought whereas Art illuminates the metaphoric wind, which blows through that structure.

Mathematics is heavily metaphorical. This is the lesson of Where Mathematics Comes From (Lakoff and Núñez). Art and math are both based on analogy-making. Meaning (semantics) in both math and art is based on analogy. There is an illusion that math is purely structural, that the interpretation of math is outside of math itself, but this is only an illusion. Math without interpretation is not interesting. Mathematicians, when actually doing math, are always working with interpretations, assigning meanings to the symbols. The formalist view of math misses completely the key role of metaphor in the human enterprise of discovering (creating, evolving) interesting mathematical truths.

Difference #4: Science reveals the body of “God” and Art reveals “God’s” mind — or is it the converse?

Math is grounded in perception (Where Mathematics Comes From), just as art is grounded in perception:

One of the great findings of cognitive science is that our ideas are shaped by our bodily experiences — not in any simpleminded one-to-one way but indirectly, through the grounding of our entire conceptual system in everyday life. The cognitive perspective forces us to ask, Is the system of mathematical ideas also grounded indirectly in bodily experiences? And if so, exactly how? — Preface of Where Mathematics Comes From

If you insist on the body-mind duality, then art and math are equally of the body or of the mind.

Difference #5: Pure Mathematics has no expression for metaphor however; it does provide us a structure that can be used for it.

Formal mathematics separates the symbolic structure of math from the interpretation of math, but the two really belong together. Math can only be interesting when it is interpreted.

Difference #6: In general, the mathematician is not interested in finding truths through nonsense as opposed to the artist who is.

Many mathematical discoveries were made by asking questions that seemed nonsensical at the time. For example, what if the parallel postulate were false?

Difference #7: The goal of art is to go beyond language. Mathematics is a language to describe what is beyond us.

Art is a form of communication between the artist and the audience. Creative art pushes the boundaries of that communication and extends the language of art. Creative math extends the language of mathematics. In both cases, language evolves, communication evolves, new metaphors evolve (are created, are discovered).

Difference #8: Artists have an insouciant tendency to get lost in their imagination. Mathematicians have an attentive tendency to map their imagination.

Mathematicians get lost in their imagination. Artists map their imagination.

Difference #9: A mathematical theory seems to come in a flash of intuition before the final product is rigorously constructed. An artistic theory seems to come much after the artwork that has been constructed in a flash of intuition.

In both cases, something rough, incomplete, and vague becomes smoother, more complete, better understood over time. Both math and art evolve. The apparent difference here is perhaps due to the ambiguity of the word theory. A closer examination of what is meant by theory may show that there is little difference between math and art in this respect.

Difference #10: Mathematical creations are not unique in the sense that they could be discovered by anyone. Artistic creations are uniquely invented by individuals.

Artistic creations are no more unique than mathematical discoveries. This difference is the myth of the hero.

Difference #11: Mathematics, among other things, is a language. Art, among other things, uses language.

The symbolic system of math is a tool for expressing metaphors. The heart of math is the metaphors. Art is the same in this respect.

Difference #12: In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite. —Paul Dirac

Poetry can tell us new things, to the same degree that science and math can tell us new things. In both cases, we can learn new metaphors, new analogies, gaining a new perspective on the world.

Alfred Ayache brought to my attention that today is Ada Lovelace Day, and that “Bloggers are asked to post about women they know and admire in technology.” I’ll list a few and you can add a few in the comments.

• Marti Hearst
• Jan Wiebe
• Claire Cardie
• Ellen Riloff
• Karen Spärck Jones
• Kathy McKeown
• …