David Kellogg On Mathematical Equations [3]

David Kellogg wrote to sys-func on 28 April 2023 at 15:27:

This kind of physics does cover Quantum Mechanics, Relativity, Maxwell, Newton, Galileo... The equations are mathematical — i.e. they can be read backwards and forwards, and they are exactly what Yaegan claimed: a different, non-linguistic, semiotic with its own anti-linguistic logic.

Blogger Comments:

[1] To be clear, this is true. In as much as an equation construes the identity of two participants, it can be 'read backwards and forwards'. However, the meanings construed in each case will be different, as demonstrated in the comments of the previous two posts.  

[2] See the comments of the previous two posts for evidence of the misunderstandings on which this conclusion is based.

From the perspective of Systemic Functional Linguistic Theory, mathematics is a field that is realised in (designed) language and (ideographic) pictorial semiotic systems. There are many reasons for interpreting mathematical equations as language rather than as a non-linguistic semiotic system. Two will be provided here.

First, mathematical equations entail a stratified content plane in which wordings realise meanings. On the SFL model, language is the only semiotic system with a stratified content plane (Halliday & Matthiessen 1999: 604):

This deconstrual of the content plane into two strata, referred to in the first chapter, is a unique feature of the post-infancy human semiotic…

The fact that the content plane of mathematical equations is stratified with a lexicogrammar is demonstrated by the fact that, unlike non-linguistic semiotic systems, they can be read aloud. On the SFL model, what is verbally projected is the wording level of content ('locutions').

Second, the only difference between

1. a² + b² = c², and
2. a squared plus b squared equals c squared, and
3. its spoken form

is the mode of expression. All three construe the same wording and the same meaning. If it is claimed that the first is not language, then someone hearing the spoken form (3) would have to ask whether it corresponded to non-language (1) or to language (2), and interpret it differently accordingly.

On the other hand, the claim that mathematical equations are 'a different, non-linguistic, semiotic with its own anti-linguistic logic' goes nowhere in terms of explanatory potential. In contrast, treating mathematical equations as linguistic explains the solving of an equation as the iterated elaboration of an identity relation, and the solution of an equation as the decoding of an unknown Token.

∞

In dismissing all the evidence that invalidates his view, and simply maintaining his original stance, Kellogg has deployed the invincible ignorance fallacy.

David Kellogg On Mathematical Equations [2]

David Kellogg wrote to sys-func on 25 April at 21:12:

Thanks, Chris--very insightful! I hadn't really parsed out voice and coding, as you can tell; I had them rather muddled, I'm afraid. The clarity of analysis is astonishing, and adds a lot of understanding.

However, the understanding I get from it only reinforces the distinction that Yaegan Doran made in his presentation. Halliday emphasises that relational clauses are never redundant. As you say, there is a difference between operative "equals" and receptive "equalled by", and there is likewise a difference between encoding"v= eE/m " and decoding ""eE/m = v". When my father writes:

(i.e. w = pressure = the Rayleigh formula, (integral of pressure term + integral of velocity term) in case the math doesn't come out in the mail....)

He is decoding and then decoding again. Not reversible.

But that's the old man doing physics, not math. When he does the math, the math itself really is redundant and reversible: "equals" and "is equalled by" really are completely equivalent, viewed as math. The encoding and the decoding, viewed as math and not as physics, are completely interchangeable. 

This to me suggests that Yaegan was right--that there is a qualitative difference between math as a semiotic system and language as a semiotic system. The former CAN be redundant, and that is the very essence of an equation. The latter is NEVER fully redundant, and that is the essence of a relational process.

Blogger Comments:

[1] This is misleading, because Halliday (1985, 1994) does not discuss relational processes in terms of redundancy. What he does say is that an identifying clause is not a tautology, and explains why this is so. Halliday (1994: 124):

In any ‘identifying’ clause, the two halves refer to the same thing; but the clause is not a tautology, so there must be some difference between them. This difference is one of form and function; or, in terms of their generalised labels in the grammar, of TOKEN and VALUE – and either can be used to identify the other.

To be clear, in the previous post, it was the introduction of the notion of asymmetry that was source of confusion, and in this post, the notion of redundancy plays that role.

[2] To be clear, it is true that the default direction of coding in a physics equation is decoding, but it is not true that this is not reversible. Such an equation becomes encoding when the Token is used to identify the Value, as when E is used to identify mc²:

Decoding

Encoding

[3] To be clear, in terms of meaning, equals and is equalled by are not completely equivalent. The operative voice of equals configures the participants as Token^Value, whereas the receptive voice of is equalled by, which is not used in mathematical formalisms, configures the participants as Value^Token. (The two also differ in choice of Subject and Theme.) What is consistent, irrespective of voice, is the identity of one participant (quantity) with another.

[4] To be clear, the direction of coding is distinct from the voice of an identifying clause. Operative and receptive clauses can be decoding or encoding. The direction of coding depends on which participant is used to identify the other: if the Value is used, then the clause is decoding; if the Token is used, then the clause is encoding.

To be clear, equations, whether used in mathematics or physics, can be decoding or encoding, though decoding is the default. However, it is not true that the two are equivalent in meaning, since they differ in which participant is used to identify the other. Again, what is consistent, irrespective of the direction of coding, is the identity of one participant (quantity) with another.

[5] See the above for evidence as to why this conclusion derives from multiple misunderstandings.

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.