Part 5 of 5. Read Part 1… Read previous part, 4… by Paul Hartal.

Chapter 5: A Magical Land of Infinite Worlds

A gentle soul and highly poetic mathematician, Georg Cantor (1845-1918), upset thoroughly the applecart of arithmetic with his Set Theory of Infinity. The mystical imagination of Cantor had created a magical land fraught with perplexing logical paradoxes that forced mathematicians to reassess the theoretical foundations of their discipline.

Infinity can be viewed as a figment of the imagination, a surreal concept transcending the human experience. Even if we could open the gate of eternity we still never would be able to reach infinity. However, Cantor treated infinity not as a verb but as a noun. Rather than approaching infinity as a process that has neither beginning nor end, he conceived it as a hierarchy of limitless sets. Some of Cantor’s infinite sets are countable because they can be put into one-to-one correspondence with each other.

Take for example the case of integers and even numbers. Common sense tells us that the entire class of integers is bigger than the thinned out series of even numbers. But in transfinite arithmetic every integer can be paired with its double and hence they have the same cardinality, which means that they are equal.

The hourglass does not show time but the trickle of sand by gravitation. Even the clock does not measure time but the pace of movement in space. So what is this mysterious thing that we call time? Is time the cosmic matrix of existence, or is it generated by it? Does time stand still, or does it flow? If it is in a state of flux, what is its speed? And since we measure speed by the ratio of travelling distance to the periodic motion of the clock, how are we supposed to measure the velocity of time? By time itself? (19, 20)

In Cantor’s enchanted world transfinite numbers form a hierarchy of infinite sets. Here the totality of integers arises as the smallest infinite cardinal number. It is followed by larger cardinalities.

Arithmetic operations with transfinite numbers yield bizarre results. Multiplying an infinite set by an integer, for example, still equals the same infinite set, but raising it to its own power produces a new transfinite number.

Cantor proved that Real Numbers (the Rational and the Irrational Numbers) are not countable. They belong to a class in a higher order of infinity than the natural numbers. He proved this by inventing the mathematical technique of diagonal analysis (17).

In Cantor’s miraculous universe of transfinite sets the part contains the whole. This stands in contrast to the prosy dominion of finite arithmetic where the whole is greater than any of its parts.

To see a world in a grain of sand
And Heaven in a wild flower
Hold infinity in the palm of your hand
And eternity in an hour.

The opening stanza in the Auguries of Innocence by William Blake (1757-1827) echoes the seemingly irrational research findings of modern mathematics and science. Yet Blake did not intend to sing the praises of these marvelous discoveries. On the contrary, Blake was a romanticist poet who revolted against the deification of reason, which he believed suffocated inspiration and vision. For his part gaining insight into a more profound reality meant trusting our instinct, imagination and energy.

Nevertheless, Blake’s rejection of reason could not prevent the triumphs of science, and ironically his poetry serves as a metaphor for the Alice in Wonderland mindscape of the electronic age. Indeed mathematical analysis and computer aided research produce new discoveries at an exponential rate. However, as Stanford physicist Leonard Susskind says, “The more we discover, the less we seem to know” (18).

Conclusion

At the end of our voyage, sailing with numbers, forms and ideas, we may say that the correlation between mathematics and nature does not have a secure and reliable base. Mathematics is a social construct; an invented world. After all, numbers do not grow on trees. But the human mind is a fascinating universe, which seems to mirror the creative forces of the physical world, of which we are an integral part. Mathematics, science and art are all invented universes, semiotic ventures to structure and interpret reality by ways of symbols (21).

We express our experience through different creative channels. Yet in spite of their specificities, mathematics and the sciences share many common denominators with poetry, painting and music.

In contrast to the widely held view of mathematics as a discipline superior to the arts, I envision their epistemological status as equal. For, the creative genius of Euler does not surpass that of Homer. Ultimately, the geometrical space of Einstein is no closer to reality than the sky of Rembrandt, and the aesthetic splendor of Cantor’s transfinite sets does not eclipse the beauty of Mozart’s symphonies.

 

16 July 2010.
© Paul Hartal, Montreal, Canada.
» Read more from Paul: ‘Portals of the Mind and the Soul.’

References (for part 5)

17. William Dunham, Journey Through Genius, New York: Wiley Science Editions, 1980, p. 252-84

18. Leonard Susskind, The Black Hole War, p. 441

19. Moses Feyngold, Special Relativity and How it Works, Weinheim, Germany: Wiley-VCH, 2008, p. 125

20. Clifford A. Pickover, Time: A Traveler’s Guide, New York: Oxford University Press, 1998, p. 96

21. Paul Z. Hartal, The Brush and the Compass: The Interface Dynamics of Art and Science, Lanham, MD: University Press of America, 1988, p.48

Complete References (for parts 1-5)

1. Eric Temple Bell, Mathematics: Queen and Servant of Science, Redmond, Washington: Tempus Books of Microsoft Press, 1989, p. 1

2.  — Mathematics, p. 21

3.  — Mathematics, p. 17

4. Albert Einstein, Ideas and Opinions, New York: Dell Publishing Co, 1954, p. 228

5. Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Communications on Pure and Applied Mathematics (1960) 13 (1): 1-14

6. Morris Kline, Mathematics: The Loss of Certainty, New York: Oxford University Press, 1982, p. 338

7. Edward Kasner and James R. Newman, Mathematics and the Imagination, Redmond, Washington: Tempus Books of Microsoft Press, 1989, p. 103-4

8. Charles Seife, Zero: The Biography of a Dangerous Idea, New York: Penguin Books, 2000, p. 1-2

9. Alden M. Hayashi, “Rough Sailing for Smart Ships: Does commercial software such as Windows NT compromise naval ship performance?” Scientific American, November 1998, p. 26

10. Ronald W. Clark, Einstein: The Life and Times, New York: Avon Books, 1972, p.271

11. John Gribbin, In Search of Schrödinger’s Cat: Quantum Physics and Reality, New York: Bantam Books, 1984

12. Leonard Susskind, The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics, New York: Little Brown, 2008, p.82

13. Brian Greene, The Fabric of the Cosmos, New York: Random House, 2005

14. David Lindley, The End of Physics: The Myth of a Unified Theory, New York: Basic Books, 1993

15. Michael Guillen, Bridges to Infinity, Los Angeles: Jeremy P. Tarcher, 1983

16. G. H. Hardy, A Mathematician’s Apology, London: Cambridge University Press, 1984, p. 84-5

17. William Dunham, Journey Through Genius, New York: Wiley Science Editions, 1980, p. 252-84

18.Leonard Susskind, The Black Hole War, p. 441

19. Moses Feyngold, Special Relativity and How it Works, Weinheim, Germany: Wiley-VCH, 2008, p. 125

20. Clifford A. Pickover, Time: A Traveler’s Guide, New York: Oxford University Press, 1998, p. 96

21. Paul Z. Hartal, The Brush and the Compass: The Interface Dynamics of Art and Science, Lanham, MD: University Press of America, 1988, p.48