A Mathematical Perspective

Of all the branches of modern science, mathematics has been the common denominator to progress and it is an appropriate starting point for our scientific discussion. However, it will be argued that there is a key distinction between a mathematical proof and a physical proof, because mathematics may only be correct within the assumptions of an original model. So, as science has moved ever further beyond the realm of human experience, the nature of empirical verification has become increasingly problematic and increasingly dependent on mathematical proof. Therefore, we need to initially reflect on this aspect of mathematics and, in part, consider the philosophical implications of how and why mathematics has come to underpin modern scientific models.

The Greek philosopher, Plato, was one of the first to articulate the relationship between mathematics and the underlying order of the universe, when he wrote of his Doctrine of Forms`. Euclid had proved that there are only five solid shapes that can be made from simple polygons, but it was Plato who went on to speculate that there were, in fact, two different universes:

• A physical universe of everyday objects.
• A mathematical universe of perfect forms.

Plato argued that there were mathematical laws governing all of nature and, as such, mathematics was not an invention of humanity, but rather the discovery of its independent existence. In fact, the corollary of Plato's argument suggests that the deeper we try to examine physical reality, the more we will become aware of a larger reality comprised of mathematical truth. Of course, at this point, you might be thinking that this idea is simply reflective of the general philosophy of the ancient Greeks. However, it seems to be an idea supported by modern mathematicians and physicists, such as Roger Penrose, Professor of Mathematics at Oxford University and Max Tegmark, a physicist at MIT. In fact, Max Tegmark is the author of a theory called the 'Mathematical Universe Hypothesis (MUH)' in which he claims that mathematical entities actually exist, which is an extension of Platonism that merely suggests that mathematical objects exist in another reality. Without recourse to too much detail, Tegmark suggestion is based on a modern scientific interpretation of physical reality:

"If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities - aliens or supercomputers, say - that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that is devoid of any human baggage like 'particle', 'observation' or other English words."

As such, he concludes that a universe of pure mathematics, free of human baggage, would  be a likely contender for the basis of our reality:

"A description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings whatsoever."

Of course, there are other schools of thought, which hold to a more pragmatic notion that mathematical concepts, like musical scores, are simply an extension of human thought. However, no matter how you align to these philosophical arguments, it is clear that any understanding of scientific development requires some complementary understanding of developments within mathematics, which is now the focus of the rest of the section.