Mathematics and Analogy

[Judith Rosen <***>]

The recent discussion comparing quantitation/digitization/mathematics with qualitation/analogy/modeling got me to thinking about what exactly mathematics is used to accomplish, in science, and WHY... Which leads me into wondering if it would be possible to come up with a set of rules about when it is productive and when it can be counterproductive to the point of being dangerous-- particularly where biological systems are concerned. This includes the fields of medicine and ecology/ecosystems science plus all of the human-created, interactive social systems (political, recreational, or otherwise). Mathematics is a complex system in its own right, a language created to communicate certain truths about human perception of aspects of the universe. It has its own set of entailment rules, which can often be used to model or illustrate entailment patterns of natural phenomena (because entailment patterns are exportable, a fact which RR said he regarded as one of the Laws of Nature.) Mathematics is often considered to be incapable of lying, beyond politics, and utterly rock solid in its capacity for representing truth. I think it might be useful to analyze why mathematics has that reputation and what, exactly, mathematics IS.

On page 4 of "Life, Itself," Robert Rosen wrote:

Let me begin with a few words about the relations existing  between the mathematical universe and the perceptual one. It is a fact of experience, for instance, that 2 sticks + 3 sticks = 5 sticks. On its face, this is a proposition about sticks. But it is not the same kind of proposition as, say "sticks burn" or "sticks float". It differs from them in that it is also about something else besides sticks, and that "something else" takes us into the world of mathematics.

The mathematical world is embodied in percepts but exists independent of them. "Truth" in the mathematical world is likewise manifested in, but independent of, any material embodiment and is thus outside of conventional perceptual categories like space and time. These facts  have, indeed, from the time of Pythagorus on, spawned another profound dualism, a dualism between idealism (which at root is an attempt to extend the reality of number to the rest of the perceptual universe) and materialism (which is an attempt to include "mathematical reality" inside conventional perceptual realms).

But of this I need not speak. To motivate our discussion, it is enough to observe that both science, the study of phenomena, and mathematics are in their different ways concerned with systems of entailment, causal entailment in the phenomenal world, inferential entailment in the mathematical. Where [qualitative and quantitative advocates] differ is precisely in their views about entailment, about what is entailed from a datum and about how that datum is itself entailed. Hence, at a sufficiently deep level, the controversy between them, and the dualism they represent, pertains to entailment itself, entailment in the abstract, free of any qualifying adjectives like "causal" or "inferential".

It is in this sense that I turn to the mathematical world in order to illuminate what it tells us about entailment. That is, I will be talking about entailment, rather than about mathematics, just as, in the example above, I could talk about number while apparently talking about sticks."

There is a lot to talk about, just in this section of the book. It seems to me that mathematics, as a system, is just as relational and interactive as the natural world is. If numbers represent the equivalent of material particles, or "things", then it's just as true that nothing happens until those numbers interact with one another via various entailment relations and according to specific organizational matters. In other words, there are semantic aspects which cannot be dispensed with, in the system called Mathematics. My father argued that all attempts to try divorcing mathematics from all semantic aspects (reducing mathematics to syntax, alone) have failed rather miserably and he even explains why this is so: Mathematics is a complex system. To reduce it to syntax is to oversimplify to the point that the system has been irrevocably changed. He uses the example of David Hilbert and his attempts to formalize Number Theory (to reduce the system to a representation in syntax, alone), which created all sorts of terrible paradoxes-- the entailments had, of course, been altered by the simplification process. That's inevitable; the organization of the system had been changed. Robert Rosen argued that the system had been changed from one that was complex to one that is simple.  (It was his view that all formalizations are simple systems and, therefore, any system which can be reduced to formalisms without loss of information is likewise simple.)

There is a further relational issue here, that of the human mind-- where the complex system of mathematics actually exists and which generated the system/language in the first place. So, not only is mathematics based on the entailments inherent in relational interactivity, within itself as a system, it exists in an "environment" on which it depends for everything including its very existence, just like any other complex system. The interaction of mathematics (the system) with the human mind (the "natural" or evolutionary environment of the system) is essential for the existence of mathematics as a system.

The fact that these "truths" can be equally illustrative of any complex system is a proof of the exportability of entailment patterns, in general: That identical entailment patterns can exist in two or more completely different systems made up of entirely different "stuff".

Comments from the group?

Judith

Web address: http://www.rosen-enterprises.com
BioTheory: An electronic journal of general science based on the Relational (Rosennean) Complexity Paradigm