Web resources on Roman Jakobson include an entry (with a lovely quote from Ilya Ehrenburg) by James Heartfield and a page (nice photo) by Raymond Bélanger. A short summary of his ideas on metonymy and metaphor is given by Colin Moock in the context of an analysis of the World Wide Web in terms of similarity disorder. The work of Jakobson's referred to here is ``Two Aspects of Language and Two Types of Aphasic Disturbances'' published in R. Jakobson and M. Halle,

*Metonymy*in the general sense characterizes the use of the part for the whole (for example ``nice set of wheels'' for ``nice car'') or vice-versa (``the administration has honored us with a visit'' = ``the principal just walked in''). In the modern critical vocabulary (see for example Simpkins on Thurber), the sense has been further enlarged to encompass any substitution based on any kind of contextual contiguity.*Metaphor*in the general sense characterizes the substitution of one similar concept for another. The similarity may be explicit (``she walks in beauty like the night ...'') or implicit (``some pig left dirty dishes in the sink'').

Metonymy (in its broadest sense) and metaphor are the principal linguistic mechanisms of humor. The joke

has two acoustic metonymies and one mock-metaphor.

Metonymy and metaphor are not just important as
rhetorical devices.
The renowned literary critic and linguistics scholar Roman Jakobson
made the point that
they correspond to two fundamental,
and fundamentally different, modes of processing symbolic information.
The symbolic system most of us are most
familiar with is a human language; these two poles (as Jakobson
describes them) are so important that they show up in speech disorders:
*contiguity disorder* is a type of aphasia in which metaphor takes over
completely. A patient will say ``spyglass'' for ``microscope'' and
``fire'' for ``gaslight'' (these are Jakobson's examples); the context-driven
grammar of the sentence disintegrates, leaving ``a heap of words.''
The polar pathology is *similarity disorder*, in which
metonymy rules. These patients say ``fork'' for ``knife'', ``smoke'' for
``pipe'' and ``eat'' for ``toaster.'' The grammatical structure
of the sentence is intact, especially from word to word; the subject
is often omitted.
In both disorders, communication is impossible.

Mathematics is also a symbolic system, devised by the same humans who developed natural languages. So it is not surprising that metonymy and metaphor should emerge as poles in the development of mathematics, both culturally and in any individual, and that awareness of this duality can be useful in learning and in teaching mathematics. This column will explore some mathematical examples of these two fundamental processes of human thought.

--*Tony Phillips
SUNY at Stony Brook*

- 1. Metonymy and metaphor
- 2. How to recognize
mathematical metonymy
- 3. How to recognize
a mathematical metaphor
- 4. Metaphors from modern mathematics:

I. Fourier analysis and the dot-product - 5. Metaphors from modern mathematics:

II. The Jordan normal form and the structure of abelian groups

© copyright 1999, American Mathematical Society.