Equations, images, understanding?

In previous posts, I have touched on understanding, and the complementary nature of conscious human understanding and the more opaque, to us at least, understanding produced by AI systems. Such systems, particularly those described as deep-learning, produce an ‘understanding’ of large and complex data sets, but without employing the kind of concepts on which humans rely for understanding.

An example of the latter is given in a 2021 review of methods for predicting chemical reactivity. Understanding of reactivity begins with a knowledge of whether the reaction is governed by thermodynamic factors (the most stable product is formed) or by kinetic factors (the product with the lowest barrier in the rate-determining step is formed). For an expert chemist, this knowledge is fundamental to any understanding of a particular reaction. But machine learning systems can achieve an expert’s ability to predict reactivity without taking these concepts of reaction mechanism into account in any way.

How might we go about attaining a human understanding from what machine learning produces?

An article in Quanta Magazine describes so-called “machine scientist algorithms”, which mutate equations billions of times to try to get a good fit to a large data set; examples are symbolic regression and sparse regression, which identify relationships in complicated data sets, but report their findings in a format human researchers can understand: a short equation. In this respect, such systems are unlike neural networks, which are highly effective, but the way they reach their results is entirely opaque to humans.

So equations are a good way to help people understand things? Not necessarily. If we think back to James Clerk Maxwell’s introduction of the modern theory of electromagnetism in the nineteenth century, its abstract mathematical formalism, very different from the mechanical models that had come before, was by no means seen as an aid to understanding. As Graham Farmelo puts it in his book The universe speaks in numbers: how modern maths reveals nature’s deepest secrets (pages 35-36):

“In 1865, Maxwell presented an improved version of this theory that he described as ‘great guns’. It did not refer to any mechanical model of the ether – there was not a vortex or an idle wheel in sight. Rather, the theory was based on differential equations that describe the rate of change of electric and magnetic fields, and of related quantities, in a way that experimenters could check. For Maxwell’s friend William Thomson, generally regarded as Britain’s most accomplished mathematical physicist, this version was a backward step, and he could never quite forgive Maxwell for taking it. Thomson did not like theories of nature to be set out purely in terms of dry and almost incomprehensible mathematics – he wanted them to be grounded in mechanisms that he could imagine in his mind’s eye.”

Mechanical models, with wheels, cogs, springs, colliding spheres, and so on were certainly a popular way of understanding scientific concepts without equations at that time, perhaps in tune with the Victorian world of steam and mechanical devices. Less so now, perhaps, when we turn more naturally to images and video. As Jenann Ismael says in Time: a very short introduction (page 44):

 “Cosmology lends itself to beautiful imagery, and there is an understandable public interest in what our universe looks like at the large scale. Equations take a lot of deciphering, but if you are given an image, you have a sense that you understand what a theory says.”

Images, then, are the key to understanding? It is certainly an article of faith in the visualization community that images enable more ready understanding of complex sets of data and information than numbers or words, and that images allow new insights to emerge. But there is much evidence that visualizations and images can be as misleading as any other form of information presentation, and can in some cases be a barrier to understanding. 

No good answer then? Certainly not a simple one. Understanding is a complex and subtle thing, and it is perhaps not surprising that no single form of information presentation can be said unambiguously to enhance it.

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