Monday, 20 December 2021

Mathematical Meditations

by Thomas Scarborough

I shall call this post an exploration—a survey. Is mathematics, as Galileo Galilei described it, ‘the language in which God has written the universe’? Are the numerical features of the world, in the words of the authors of the Collins Dictionary of Philosophy, Godfrey Vesey and Paul Foulkes, ‘free from the inaccuracies we meet in other fields'? Many would say yes—however, there are things which give us pause for thought.

  • Sometimes reality may be too complex for our mathematics to apply. It is impossible to calculate in advance something as simple as the trail of a snail on a wall. Stephen Hawking noted, 'Even if we do achieve a complete unified theory, we shall not be able to make detailed predictions in any but the simplest situations.' If we do try to do so, therefore, we abuse mathematics—or perhaps we should say, in many contexts, mathematics fails. 
  • Our measurement of the world may be inadequate to the task—in varying degrees. I take a ruler, and draw a line precisely 100 mm in length. But now I notice the grain of the paper, that my pencil mark is indistinct, and that the ruler's notches are crude. In many cases, mathematics is not the finest fit with the reality we deal with. In some cases, no fit at all. I measure the position of a particle, only to find that theoretical physicist Werner Heisenberg was right: I have lost its velocity. 
  • The cosmologist Rodney Holder notes that, with regard to numbers—all numbers—'a finite number of decimal places constitutes an error'. Owners of early Sinclair calculators, such as myself, viewed the propagation of errors in these devices with astonishment. While calculators are now much refined, the problem is still there, and always will be. This error, writes Holder, 'propagates so rapidly that prediction is impossible'. 
  • In 1931, the mathematician Kurt Gödel presented his incompleteness theorems. Numbers systems, he showed, have limits of provability. We cannot unite what is provable with what is true—given that what this really means is, in the words of Natalie Wolchover, ‘ill-understood’. A better known consequence of this is that no program can find all the viruses on one’s computer. Consider also that no system in itself can prove one’s own veracity. 
  • Then, it is we ourselves who decide what makes up each unit of mathematics. A unit may be one atom, one litre of water, or one summer. But it is not that simple. Albert Einstein noted that a unit 'singles out a complex from nature'. Units may represent clouds with noses, ants which fall off a wall, names which start with a 'J', and so on. How suitable are our units, in each case, for manipulation with mathematics?
  • Worst of all, there are always things which lie beyond our equations. Whenever we scope a system, in the words of philosophy professor Simon Blackburn, there is 'the selection of particular facts as the essential ones'. We must first define a system’s boundaries. We must choose what it will include and what not.  This is practically impossible, for the reason that, in the words of Thomas Berry, an Earth historian, 'nothing is completely itself without everything else'. 
  • I shall add, myself, a 'post-Gödelian' theorem. Any and every mathematical equation assumes that it represents totality. In the simple equation x + y = z, there is nothing beyond z. As human beings, we can see that many things lie outside z, but if the equation could speak, it would know nothing of it. z revolts against the world, because the equation assumes a unitary result, which treats itself as the whole.
Certainly, we can calculate things with such stunning accuracy today that we can send a probe to land on a distant planet’s moon (Titan), to send back moving pictures. We have done even more wonderful things since, with ever increasing precision. Yet still the equations occupy their own totality. Everything else is banished. At what cost?


Keith said...

I believe it’s fair to characterise mathematics as the mother tongue of the natural sciences. Mathematics is a key tool by which scientists seek to understand the universe, and everything in it. Putting a finer point on it, the theoretical physicist Eugene Wigner published a paper whose title referred to the ‘unreasonable effectiveness’ of mathematics, then explained how and why. Mathematics is thus a tool that helps to describe the natural laws and organisation of the natural world.

According to this view, mathematics seems to provide the building blocks of the universe. We shouldn’t, perhaps, describe mathematics short. An example consists of that part of the mathematics of Einstein’s theory of general relativity predicting the existence of ‘gravitational waves’, whose presence in the universe was proven empirically a century later. Nothing to sniff at.

Mathematics thus has enormous predictive capabilities, increasingly to uncover dimensions of reality. A concrete example entailed the mathematical hypothesis that a fundamental particle exists whose field is responsible for the existence of mass. The particle was theoretically predicted, in mathematical form, by physicist Peter Higgs. Existence of the particle, eponymously named the Higgs boson, was confirmed by tests at the CERN particle accelerator several decades later, resulting in the Nobel Prize in physics.

As Aristotle presciently summed it up, the ‘principles of mathematics are the principles of all things’. Aristotle’s broad stroke foreshadowed the possibility of what millennia later became known in the mathematical and science world as a ‘theory of everything’, unifying all forces, including the still-defiant unification of quantum mechanics and relativity. Though, in my opinion, any such purported ‘theory of everything’ will likely prove frustratingly incomplete, found to leave out many other somethings. But wouldn’t that state of ‘incompleteness’ describe the ‘reality of everything’ that every field, without exception, might overly boldly put forth?

Thomas O. Scarborough said...

Thank you, Keith.

I shall describe what I see in the ‘state of affairs’ of mathematics here described. You say that mathematics has enabled us not only to describe things, but to discover things, and that in astonishing ways.

I see two things in your comments which describe the 'dark side' of mathematics, a paradox of mathematics. a) We should not ‘describe mathematics short’. Yet b) even a TOE will ‘leave out many other somethings’.

This means that mathematics has stunning results, yet leaves things out. But what does it leave out? And what does that do?

Take the principal chemical formula on which the automobile was built. C8H18 + 12.5 O2 → 8 CO2 + 9 H2O. This was pure genius, but it left out reactions which now threaten all of humanity. This was not a practical error, but a problem inherent to mathematics. All mathematics ‘leaves out many other somethings’.

I consider, therefore, that both mathematics and its application creates a vast and dangerous illusion. The more we focus on our stunning results, the more damage we do.

Keith said...

Actually, Thomas, I don’t see a ‘paradox’ in the following two statements, which you (incompletely) drew from my comment above: (a) ‘We shouldn’t, perhaps, describe mathematics short’. (b) ‘Though, in my opinion, any such purported theory of everything will prove frustratingly incomplete, found to leave out many other somethings’. To my mind, the two statements are neither incompatible nor contradictory nor paradoxical.

Here’s why. Sure, mathematics flexes prodigious power in our efforts to describe — that is, acquire knowledge and understanding of — natural phenomena. Including enormous ‘predictive’ power, of the kind that foreshadowed gravitational waves and the Higgs boson. Yet, would anyone really doubt that mathematics will continue those processes indefinitely, and that theories of everything are, and always will be, chimera? No matter the field of study? Surely, there will always be more to learn, where mathematics (along with other fields of inquiry) will play a key role.

Importantly, you left out of your query my key follow-on observation: ‘But wouldn’t that state of incompleteness describe the reality of everything that every field, without exception, might overly boldly put forth?’ My point being that no discipline — from mathematics to the natural and social sciences to the whole range of humanities and human study — there will never be a true, end-of-all-inquiry ‘theory of everything’. There will always be more to learn as we seek a better understanding of ourselves and the universe in which we live. No field(!) is exempt from that state of affairs.

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