David Kellogg On Mathematical Equations [1]

David Kellogg wrote to sys-func on 24 April 2023 at 12:00:

I remember being vaguely dissatisfied with a presentation that Yaegon gave on this topic at University of Sydney, simply because he over-stressed the difference between mathematics and spoken language and at one or two points even argued for distinct origins in different kinds of logic. But now I see his point.

The terms "coding" and "decoding" imply asymmetry, and in fact asymmetry is the sine qua non of Halliday's understanding of relational processes. Understanding the grammar of a statement like "Tom is the tall one" (value-to-token, or encoding) or "Tom is the treasurer" (decoding) is really understanding that the sentence is NOT redundant — that the expressions are quite different when you reverse them. Encoding and decoding sentences are no more reversible than part-whole relationships, e.g. "All donkeys are animals" and "all animals are donkeys".  

But mathematical equations do not in themselves imply this asymmetry. For example, your equation could be reversed with very little or no difference in meaning,  "All donkeys are animals" is true and "all animals are donkeys" is false. But "c squared equals a squared plus b squared" is just as true and just as false as "a squared plus b squared equals c squared".

I was just discussing this with my father, who has been working on some data from the Parker Solar Probe, now entering the sun's corona. He wants to demonstrate that the heating of the solar wind  is caused (or at least could be caused) by the damping of ion acoustic waves in the solar wind. He pointed out to me that his papers are usually written in a mixture of mathematics and English. Here, for example, he is discussing how to determine the motion of a particle in an electric field. He says::

"dv/dt  = eE/m   

"So that

"v= eE/m "

Because the equations are embedded in language, they are not reversible. But the non-reversibility (that is, the fact that they are decoding relationships, with the token on the left and the value on the right) is an artefact of the language and not a logical consequence of the mathematics, just as the order of "a squared plus b squared equals c squared" is an artefact of our alphabet. 

 

Blogger Comments:

[1] This is misleading, because Halliday (1985, 1994) does not discuss relational processes in terms of asymmetry. What Kellogg refers to as 'asymmetry' is the functional difference between the two participants of an identifying process: Token vs Value and Identified vs Identifier. In a decoding clause, the asymmetry is Token/Identified vs Value/Identifier, whereas in an encoding clause, the asymmetry is Token/Identifier vs Value/Identified.

[2] To be clear, understanding the grammar of identifying clauses is understanding the functions that each clause constituent realises: Token or Value, Identifier or Identified.

[3] To be clear, encoding and decoding are reversible, but changing the direction of coding changes the meaning that the identity construes: Value encoded by reference to Token vs Token decoded by reference to Value.

[4] To be clear, part-whole relationships are reversible, in terms of coding, when they are construed as identifying rather than attributive. For example, the sports centre comprises four buildings is decoding in contexts where four buildings identifies the sport centre, and encoding in contexts where the sport centre identifies four buildings.

[5] To be clear, attributive clauses are not reversible in terms of the direction of coding because coding is restricted to construals of identity, whereas attributive clauses construe class membership (Halliday & Matthiessen 1999: 145-6), not an identity. (Trivially, reversing the participants in all donkeys are animals yields animals are all donkeys.)

[6] This is misleading, because it is untrue. If the participants in the clause a squared plus b squared equals c squared are reversed, all that is preserved is the identity of a squared plus b squared and c squared. The reversal changes not only what serves as Token and Value, but what serves as Theme and Rheme and what serves as Subject and Complement. Moreover, it can also involve a reversal the direction of coding:

[7] To be clear, the equations are reversible, though the reversal changes the functional role served by each nominal group in the equation, as explained in [6].

[8] To be clear, what makes these equations decoding is the conflation of Value with Identifier, not the sequence of Token and Value. If the Token on the left is used to identify the Value on the right, these equations are encoding.

[9] To be clear, the ordering Token^Value is given by the clause being operative in voice. But this is insufficient in itself to make the clause decoding; see [8]. 

Importantly, the mathematical equation uses the operative identifying clause as its template, and solving an equation involves repeatedly elaborating the equation, while preserving the identity relation, until each unknown is construed as a Token that is decoded by a Value.

[10] To be clear, this is the Pythagorean equation. The letters a and b stand for the two sides that meet at a right angle, and c stands for the hypotenuse.