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The Unreasonable Effectiveness of Mathematics in Science and Engineering


Why is mathematics - a product of the mind - so outstandingly successful at predicting the behavior of physical systems?

Einstein is said to have remarked that "The most incomprehensible thing about the universe is that it is comprehensible."  This observation was elaborated by Eugene Wigner in his famous paper in Pure Mathematics (Volume 13, Number 1, February 1960) entitled 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'

The theme was further developed in  'The Unreasonable Effectiveness of Mathematics' by  R. W. Hamming in The American Mathematical Monthly (Volume 87, Number 2, 1980), which considered the predictive, as well as descriptive powers, of mathematics in relation to engineering.

Two surprising conclusions appear from these papers:

(1) Although it is a product of the human mind, mathematics is also involved in some strange metaphysical way at the deepest levels of physical existence. To quote Wigner:

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

(2) There is no Darwinian explanation for the presence of mathematical abilities within the mind. The ability to understand physics could not have arisen by evolution. Although our bodies may well be the product of random mutation and selection al the way from amoeba to man, our minds have some 'unevolved' dimension.  To quote Hamming:

"But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require".

"If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics."

Or Wigner again:

"Certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess."

Note - this should not be taken as evidence for Genesis, Special Creation, Young Earth, Flat Earth and all the rest of the creationist mumbo-jumbo. It does, however point to aspects of the mind which have not arisen out of the purely physico-chemical processes of evolution.  (The process of evolution can in fact be regarded as an algorithm. Evolution produces complex structures out of the endless algorithmic loop of replication, random error in copying, and resources which are insufficient  to allow all replicated copies to survive to replicate themselves in turn. Evolution is the algorithm, and the DNA codes of living organisms are the corresponding datastructures which are modified)

So we are left with something of a mystery. According to the physicalist worldview, the mind (including mathematicians' minds) is an epiphenomenon of matter which has evolved solely to ensure the survival of the selfish genes which code for it. Why should this 'top-level' phenomenon have such intimate access to the 'bottom level' phenomena such as quantum physics? After all, the two levels are supposedly separated by less well-understood (in some cases) explanatory layers such as evolutionary psychology,   neurology, cell biology, genetics, molecular biology, and chemistry.

Possibly a combination of Wheeler's Participatory Anthropic Principle and Von Neumann's theory of the origins of number   may offer a glimpse of the way forward.  Suspend disbelief for the moment and consider the possibility of the existence of a disembodied mind or minds.

Consider also that the universe began as a quantum event which remained in a superposed state of all possibilities until acted upon by the mind of an observer (Copenhagen interpretation of quantum mechanics).

Now an observer existing in the absence of any objects or forms will have no knowledge of any material things. She will, however, have knowledge of mathematics, as mathematics arises from mind comtemplating emptiness, and needs no material objects or things to count.

A mind whose prior experience had been the contemplation of emptiness, might upon observing a superposed quantum system, have a predilection for collapsing the superposition around a primordial mathematical framework.

The mind of the sentient being who first collapsed the superposition is not different in fundamental nature to the minds of the physicists who observe the results  -   Tat Tvam Asi.

- Sean Robsville

See also:

Emergence
The mind cannot be an emergent property of the brain or any other physical system, since emergent properties and emergent phenomena are psychological in origin, and require the pre-existence of an observer's mind in order to become manifest.

Participatory Anthropic Principle
'....Throughout its early history the universe continued to develop as an immense superposition of probabilities. Not only was the structure of the universe superposed, but all logically possible states of matter, physical constants, properties and laws were simultaneously present and evolving into ever increasing diversity. Quantum theory states that any physical system remains in a superposed state of all possibilities until it interacts with the mind of an observer. At some instant one of these possible alternative universes produced an observing lifeform - an animal with a nervous system which was sufficiently evolved to form a symbiotic association with a primordial mind. The first act of observation by this mind caused the entire superposed multiverse to collapse immediately into one of its numerous alternatives...'

Von Neumann and the empty set
'.....in the final analysis the entire number system has been generated by the play of mind on emptiness, in the complete absence of the need to refer to any material thing, or things, which are being counted.  Numbers do not exist by reference to physical reality, nor are they self-existent, abstract 'things in themselves'...'


Copenhagen interpretation
'...It is important to emphasise that the mathematical equations of quantum physics do not describe actual existence - they predict the potential for existence. Working out the equations of quantum mechanics for a system composed of fundamental particles produces a range of potential locations, values and attributes of the particles which evolve and change with time. But for any system only one of these potential states can become real, and - this is the revolutionary finding of quantum physics - what forces the range of the potentials to assume one value is the act of observation..'

Sunyata - the emptiness of all things
'.......all things have no fixed identity ('inherent existence') and are are in a state of impermanence - change and flux - constantly becoming and decaying. Not only are all things constantly changing, but if we analyse any phenomenon in enough detail we come to the conclusion that it is ultimately unfindable, and exists purely by definitions in terms of other things - and one of those other things is always the mind which generates those definitions...'

Non-algorithmic phenomena
'The great difficulty in talking about non-algorithmic phenomena is that although we can say what it is in general terms that they do, it is impossible by their very nature to describe how they do it.  (If we could describe in a stepwise manner what was going on then we could write a computer program to perform the task).'

Mathematics: The Bridge to an Integral Science of Experience
The scientific method is generally thought to require that scientific verification draw only from outer experience, thus excluding all inner experience as a valid basis for scientific verification. Consequently, the scientific study of consciousness today is largely limited to the physical sciences and the externally observable correlates of consciousness. Mathematics, however, provides an example of a rigorous science based on inner experience that is nonetheless verifiable. Contrary to widespread belief, it is possible, therefore, for an authentic science to be based upon inner experience.

Buddhist Teachings
on the mind, personal relationships, meditation and the spiritual path.

The Three Poisons in Buddhism
Rage, Greed and Irrationalism

Arguments against Buddhism
- the best way to understand the strengths of a philosophy is to attempt to refute it!

RATIONAL BUDDHISM
If we regard Buddhism as a combination of a philosophy, psychology and religion, then how much mileage can we get from the first two aspects before we have to start invoking religious faith?

Christian versus Buddhist worldviews

Buddhism in Everyday Life
The Daily Meditation