You might be pleased to read that for this post, for this one delicious interval, I will not be writing about religious beliefs.
To continue my previous comments about whom we might deem to be a ‘scientist’, I wish to direct your attention to a different, and far less obvious opposition.
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As a reminder, here is a brief summary of my attempt at, what I deemed to be, a ‘pattern-seekers manifesto’:
- It is our collaborative goal to search for patterns in our cosmos.
- These guesses attempt to gain greater predictive power of our environment, and, with this facility, we are able to control our future so to better and enrich our lives.
- We believe the greatest progress of our capabilities can be achieved through an ongoing process of bold conjectures and relentless refutations. This way, we can quickly eliminate false ideas before they cause any harm.
Opposition to this scientific inquiry can be twofold.
Most noticeable are challenges in the form of forceful restraint, enforced by those who hold religious, or other ideological beliefs.
Secondly, and to which I focused the attention of the previous post, it can take the form of a popular belief among those otherwise unrestrained individuals working in the sciences.
For the latter consideration, the personal beliefs of, say, a biologist, or a geologist can be antithetical to the program described above. Religious belief in miracles is a widespread example of such an idea that is contrary to the goal of searching for patterns. Indeed, it is a flat out abandonment of the assumption that we can invent universal laws to describe our universe.
And it’s no use pretending that one or two miracles, usually claimed to have occurred in our distant and ignorant past, can be accepted by the mind of a scientist.
Grant a single miracle, and you disregard the pattern-seeking methodology, and pretend to know when and where it is unachievable.
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Alright, I might have mentioned religious belief once or twice.
Before I move away from it, notice that the defining impact of such ideas on scientific investigation is usually to discourage or discredit pattern-seeking, in favor of some supposedly ‘revealed’ information.
In contrast, the non-religious idea of ‘mathematical Platonism‘ has the opposite effect of over-reaching – claiming to find patterns that are beyond our capabilities.
In short, Mathematical Platonism is the assumption that mathematics is ‘real’ (whatever that may mean), and that it can be called upon as an explainer of mysterious problems.
The idea is not obviously antagonist to the scientific program. Hopefully, a few examples will illustrate the trouble.
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The great astronomer Johannes Kepler is most frequently mentioned as the discoverer of three ‘laws’ that describe the motion of the planets around the sun.
They are perhaps the first examples of scientific theories, as we would understand the term today.
Bearing his name, these numbered laws are as follows:
- The planets move around the sun in ellipses with the sun at one of its foci.
- The area swept out by a planet’s orbit is proportional to the time period.
- The square of the time period of a planet’s orbit is proportional to the cube of its orbital radius.
What is less known is that Kepler arrived at these laws after proposing an very different type of pattern.
In his 1596 work ‘Mysterium Cosmographicum‘, Kepler thought that he could ‘explain’ the ratios of the planets’ orbital radii with reference to the ‘five Platonic Solids‘. I’ll omit the details for now, and perhaps describe his ideas in another article.
Rather, I would like to draw to your attention to Kepler’s methodological disposition. He considered it scientifically relevant to form patterns between mathematical objects (the Platonic Solids) and physical variables (the orbital ratios of the planets).
This original scheme was abandoned when Kepler obtained more precise astronomical data. The new information enabled him to discover that planets actually move in ellipses – not perfect circles as previously assumed.
However, factual refutations, no matter how decisive, are incapable of shaking a methodological assumption. Kepler remained a mathematical Platonist the rest of his life, seeking modifications to his original thesis ever time it encountered a new contradiction.[1]
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Since learning this, I have considered what our scientific history might have been like if Kepler was correct. Not methodically correct – I mean if his pattern actually fitted the facts.
Imagine a solar system in some distant part of the galaxy where their orbital radii are, within the precision of their measurements, exactly as Kepler originally described.
Or perhaps the orbital radii conformed to so other mathematical sequence – prime numbers or the Fibonacci sequence, as examples.
I wonder how this civilisation might come to abandon such a theory, formed by Kepler’s equivalent.
In isolation, the theory would continue to be unfalsified. How would this impact their scientific progress? Would it hinder their equivalent Newton from finding a universal law of gravitation? Would it stultify their technological advance?
I would conjecture that it would substantially retard their understanding of astronomy and cosmology.
Perhaps, eventually, after discovering much more about their own world, they’d come to see this pattern as methodologically dubious, and dismiss it on philosophical grounds.
They might first ask why the Platonic Solids need to be arranged in a seemingly arbitrary way? One set of numbers (the ratios of the orbital radii) have been replaced with an equally mysterious set of numbers – the ordering of the Platonic solids.
Finally, they might come to realize that this type of pattern has no physical significance.
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Kepler’s mistake was to form a pattern between mathematical objects and a physical variable. In modern physics, we compare physical variables with each other.
Mathematics is not a comparatee, but the comparator. It is the systematic classification of patterns, and we use this system to describe patterns in the physical universe.
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Before moving to the next example (in the next post), I’d like to end with a brief comparison.
Before the quantum mechanics of the atom was understood, many patterns emerged that described the light emitted from particular atoms.
For example, there was the ‘Balmer Series‘ for the hydrogen atom which related the sequence of emitted wavelengths to a numerical sequence.
Later, the pattern was generalised to the ‘Rydberg Series‘, which was could describe the ‘emission spectra’ of any atom.
These two equations, if taken too seriously, suffer from the same mathematical Platonism as Kepler’s idea. Namely, they purport to ‘explain’ a sequence of a physical variable (wavelengths of light) with a mathematical construction (a numerical series).
However, in this case, the venture was not fruitless.
Neils Bohr was able to see that the numerical sequences, and the sequences of wavelengths, were also related to the orbital radii (and potential energies) of electron orbits in atoms.
Eventually, Bohr demonstrated that this sequence could be further equated with an expression involving some physical variables of quantum mechanics, thus eliminating the numerological crutch.
The reason there was no one to do this for Kepler is to do with a difference in scope. Kepler was only investigating our solar system. Of course, at the time, it was believed to be the only one in the universe.
Once you realize that our solar system is just one of many, which vary in number of planets and orbital radii, then it is clear that Kepler’s theory cannot be shaped into a universal law. In contrast, the orbital radii of electrons are thought to be universal - the same for all atoms of the same element.
In addition, we have learnt that these planetary systems have their origins in clouds of dust, which in turn are dependent on the initial conditions of the universe.
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Needless to say, these considerations are all in hindsight.
And this is part of the problem with mathematical Platonism. Once you allow mathematical patterns to be compared to physical patterns, it is very difficult to determine their significance. It can lead to the helpful first steps to physical theories, or, most of the time, to useless numerology.
It is only when we compare physical variables to one another that we can hope to discover patterns that have objective significance.
References
[1] A very clear biography of Kepler has been provided by:
Koestler, Arthur (1959) ‘The Sleepwalkers (The Watershed)’
