Pythonism

code and the oracular

Mathematics – Unity in Diversity

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One of the things about math is that the more problems we look at the more we find a finite set of mathematical objects that can be generated in more than one way, observed in more than one context. These patterns recur in varying and apparently unconnected places.

Here are just a few of the “great ideas” and their specialised manifestations. Every area of maths and science displays them. Patterns that appear in separate domains, systems, phenomena and problems. I say apparently unconnected because, if anything, writing this article has made me realise how what the mystics say is so true: all is one and interconnected.

This list is of course vastly incomplete. After a while of coming up with more and more patterns you just say “right, that’s enough to make the point!” In fact I believe the one true exhaustive list of this kind is infinite. I have no idea for a proof or disproof of that ! Suggestions please.

pi

  • found in area or circumference of stars, planets etc
  • Euler’s formula e^ipi+1=0
  • many continued fractions, products and sums
  • Gaussian Integral
  • Stirling’s approximation
  • cosmological constant
  • Heisenberg’s Uncertainty Principle
  • Einstein’s field equation of general relativity
  • Coulomb’s law for the electric force
  • everywhere in the universe

the exponential function

  • growth of population
  • advance of technology
  • explosives including atomic bombs
  • pyramid schemes in business
  • acoustic feedback
  • compound interest in finance
  • anything showing invariance under differentiation

the integral and the derivative

  • motion
  • curvature
  • energy
  • temporality
  • fields and forces

complex numbers

  • electronics and signal processing
  • electromagnetism
  • many other quantum phenomena
  • chaos theory
  • Lie algebra and spacetime topology

graphs and trees

  • electronic networks
  • social connections
  • road systems
  • molecular structure
  • blood vessels
  • river deltas
  • plant structure
  • composition of probability outcomes
  • heredity and family trees
  • neural structure

the group

  • permutations
  • particle symmetries
  • molecular symmetries
  • handedness in nature
  • crystallography
  • rubik’s cube and other puzzles
  • cryptography and codes

recursion

  • Godel’s theorem
  • Russell’s and other paradoxes
  • many algorithms eg factorial functions
  • self awareness and consciousness

self similarity

  • self similarity in natural forms
  • optimum fractal antenna designs
  • hausdorff dimension
  • ferns and other plants
  • coastlines and geology
  • lightning

the spiral

  • galactic structure
  • sea shells
  • hair on peoples’ heads
  • helical molecules (eg DNA)
  • sunflower heads and other plants
  • vortices in fluids (the plughole syndrome)

the binomial numbers

  • many combinatorial systems
  • card hands
  • pascal’s triangle
  • expansion of (x+a)^n

The Normal Distribution

  • in statistical analysis of millions of natural systems
  • IQ in humans
  • velocity of molecules in a gas
  • Probability density function in quantum systems

power laws

  • frequencies of words in languages
  • frequencies of family names
  • sizes of craters on the moon
  • solar flares
  • sizes of power outages, earthquakes, and wars
  • popularity of books and music

trigonometric functions: sin, cos, tan

  • waveforms of all radiation
  • surfing
  • simple harmonic motion in pendulum or any oscillator
  • sums of certain infinite sequences
  • properties of triangles and space
  • resultant forces and dynamics

infinity

  • spacetime curvature in black holes
  • sums of divergent sequences
  • cardinality of certain sets
  • Eternity
  • infinitesimals and limits

I would like to ask: are these patterns somehow “more real” than ordinary “stuff” because they occur more than once in different contexts? Generalisation leads back to a smaller set of “God’s favourite themes” – leitmotifs which underly and describe natural and numeric structures. This is the essence of math.

Was Plato right ? All roads lead to Rome. In other words many different problems have the same structures governing their solutions, these structures are unvarying and lie behind and beneath everything in the universe. They are ideal forms which somehow recur. Are they an order of reality higher than and above the ordinary world of phenomena ? I can see no better justification for studying math than this incredible consistency and regularity in the universe. Even if you never find God there are miracles to be found…The alternative to this view is more sociological and says that these ideas are merely social constructs, kinds of myths, that emerge from a human language game. I’d like to know how many mathematicians believe this, it may be that it’s just sociologists !!

Of course as well as describing so many patterns in nature these fundamental objects also relate to each other in profound ways. Pick any two of the above patterns and you will see a host of equations and relationships connecting them. In fact just examining this principle of “all roads lead to Rome” we are actually looking at the whole of mathematics.

This is what mathematics is: a doorway into a higher reality populated by archetypal forms that underly everything that exists. I feel so lucky.

However a note of caution: now we see the absolute power that mathematics confers it seems that with this much knowledge any mathematician may find they struggle with a superiority complex. Humility is part of the game too though ! With humility we avoid becoming brittle and overconfident. With humility we still admit the openness of not knowing what’s going to happen next. With humility we see that the universe doesn’t just go Pop! and disappear to be replaced by a book of all math. With humility we don’t suggest that the humanities are empty. Overconfidence makes us miss the aha! or Zen solution to problems and think we have encapsulated something that is infinitely beyond encapsulation: REALITY !

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Written by Luke Dunn

April 14, 2010 at 4:02 pm

One Response

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  1. You probably can add

    (1) i^i = exp(-pi/2) …… the funny thing is (complex)^(complex) = real ! …… weird but mathematics justifies it !

    (2) Galois Theory, it is probably one of the two most beautiful and elegant theories I have read. It relates Geometry to Groups to Solutions of Equations ( There is a book, ‘Abel’s Proof’, which is a wonderful discussion on this topic)

    Further, you would have noticed that these lists that you have made are not really independent ….but are rather interconnected (more mesh like).

    Arkapravo

    May 15, 2010 at 5:22 am


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